Alchemist has a powerful skillset that benefits him in a few different roles.
Q - Acid Spray
Sprays high-pressure acid across a target area. Enemy units who step across the contaminated terrain take mixed damage per second and have their armor reduced.
Damage Per Second: 14/20/26/32
Armor Reduction: 3/4/5/6
Cooldown 22 Mana 130/140/150/160
Acid Spray is a powerful AOE spell that has many purposes. It's an armor shred, an AOE damage spell, a farming tool, and a lane control tool. The mana cost does scale and is pretty steep at all levels, so be careful of your mana so that you have enough for you other spells!
W - Unstable Concoction
Alchemist brews up an unstable concoction that he can throw at an enemy hero, to stun and deal damage in an area around the explosion. The longer the concoction brews, the more damage it deals and the longer the stun. After 5 seconds, the brew reaches its maximum damage and stun time. However, after 5.5 seconds, the concoction will explode on Alchemist himself if not thrown.
Maximum Stun Duration: 1.75/2.5/3.25/4
Maximum Damage: 150/220/290/360
Cooldown 16 Mana 120
SUB ABILITY - Throw Unstable Concoction
Unstable Concoction can be your biggest tool in your arsenal or your biggest bane. Ever since the nerfs to the stun in 6.80, it's much easier to backfire. This makes support and ganker Alchemist much less viable. However, more experienced players can work around this nerf and still pull off the maximum damage without stunning themselves. Sometimes, it's better to play it safe. Sometimes instead of getting that 360 damage 4 second stun, sometimes it's safer to just make that 220 damage, 2 second stun, especially against slippery opponents like Morphling, Weaver, and Bounty Hunter.
E - Greevil's Greed
Alchemist gets bonus gold for killing creeps; the bonus stacks if the creeps are killed within a period of time, after which the count is reset. The bonus gold is capped at 30 gold.
Bonus Gold Per Kill: 1/2/3/4
Base Bonus Gold: 4/6/8/10
This is what makes Alchemist such a great farmer. The more last hits he gets, the more bonus gold he gets. It's very much possible to have a 1000+ GPM on this hero with the right items. However lucrative it maybe be to max this skill first, avoid doing so. It's much better to max your stun.
R - Chemical Rage
Alchemist causes his Ogre to enter a chemically induced rage, reducing base attack cooldown and increasing movement speed and regeneration.
Base Attack Time: 1.4/1.2/1.0
Bonus Health Regeneration: 50/75/100
Bonus Mana Regeneration (Base Mana Regeneration): 3/7.5/12
Bonus Move Speed: 30/40/60
Cooldown 45 Mana 50/100/150
Chemical Rage is what makes Alchemist a powerful hero. A little bit on mechanics first, however. Every hero has a base attack time; their attack's rate without any attack speed. The lower the BAT, the faster you attack essentially. Think of it as a car. A high BAT is like a really old, not very aerodynamic car. You can put a big engine into it (a lot of attack speed), but it's not going to be as fast as a newer, state of the art vehicle. Alchemist's ultimate gives him a lower BAT (the lowest in the game. For comparison, the second highest BAT in the game right now is Anti-Mage with 1.45 and the average being 1.6), meaning he benefits very well from attack speed items. As a carry, the most important thing is the BAT, which will increase your DPS and is what you build your hero around. As a ganker, the most important thing is the regen and movespeed, which allow you to roam a lot better. With a duration of 25 seconds and a cooldown of 45, you are at your maximum potential once every 20 seconds.
Item Builds and Justification
For both builds I recommend not going by a set starting item build. Depending on your lane, you should grab different items. If you're rushing bottle, a Stout Shield+Salve will give your an extra 250 gold toward your bottle. If you're expecting free farm, a Quelling Blade is nice to have. If expecting a hard lane, extra regen or grabbing a Stout Shield is recommended.
The same goes for all heroes. Look at the enemy lineup, your lineup, and what you expect your lanes will be. In the case of Alchemist mid, your team has you, a Carry Sven, jungle Chen, offlane Bounty Hunter, and a support Crystal Maiden. The enemy lineup has a Pudge, a Weaver, a Kunkka, a Bane, and a Lina. Judging by their lineup, it could be a lot of things. It could be a trilane Kunkka+Lina+Bane, it could be dual lanes with Weaver+Lina and Kunkka+Bane. Or it could be a 2 man ganking combo with either Bane+Pudge, Lina+Bane, or Kunkka+Bane. However, chances are you will get a Pudge at mid. Because of Chemical Rage, he probably can't kill you, and being that you are both melee, he likely can't harass you very well (although you may trade blows.) Judging by this, grabbing a Quelling Blade, two Tangoes, and branches will be your best bet. If it turns out to be a Kunkka or Weaver mid, you have enough regen to stay in lane.
Early Game Items
Early game item's don't differentiate much between the builds. The carry typically doesn't focus on early game items and will only grab a couple of really cheap ones in order to help in lane.
The above is an example of my preferred early game Carry build. The Wand and Soul ring are very cheap, which doesn't set you back on your big items much. The Wand is for when you get ganked, as a burst of HP can save your life. The mana is for when you're stacking camps, as Alchemist has subpar INT gain (but very good starting INT) and Acid Spray's mana cost is massive.
The above is an example of my preferred early game Ganker build. The regen from Bottle and Medallion make sure you always have mana to cast your spells. Bottle helps for rune control and can also give mana for stacking camps. Medallion's armor shred stacks with your already existing Acid Spray and since Concoction is physical damage, it will amplify the damage your stun does dramatically (sometimes doing 100% damage or more depending on the hero!). The armor helps solve Alchemist's poor agility and base armor. Soul ring is also good for a ganker too, however since you'll be in fights more often, you might find yourself a bit lower on HP than usual. A bracer is also a nice and viable item to build as it builds into a Drum, a strong item on a Ganking Alchemist.
Core items are where the Ganker and Carry build differ greatly. The Carry build is more expensive but gives bigger, better bonuses and is more oriented around making use of Alchemist's ultimate. The Ganker build is much cheaper and is more oriented about positioning to land your stun while also giving you cheap stats to make you more threatening early on.
Phase Boots are Alchemist's boots of choice as they give him good mobility and allow his to chase down faster targets in order to land his stun. If you are confident in landing your stun, going for Power Treads is powerful because of the bonus attack speed and stat swapping. Arcane boots are also viable if you're playing more of a support role.
Blink Dagger is the god of all ganking items. For a cheap 2150 gold, you can charge up your stun out of the fog, Blink in with 2 seconds on the timer, Medalion, Spray, and hit them with the stun. It's also good for chasing. If you're about to do a second stun, charge your stun and when the enemy thinks they've gotten away from you, make them **** their pants by blinking right in their face and stunning them.
Drum of Endurance is a cheap item which gives a little bit of everything Alchemist needs. He needs HP for his poor stat gain, he needs a little bit of mana to allow him to roam, he needs some attack speed in order to make use of his ultimate, and he needs some mobility in order to help land his stun. This gives him a little bit of everything he needs and is a fantastic, cheap buildup from a Bracer.
As I said earlier, Phase boots allow Alchemist to land his stun and chase. With the bonus movespeed from Chemical Rage, it'll be easy to chase and run down enemies. Treads are still viable on this build as well if you're thinking about stacking straight attack speed.
Mjollnir is your one and only farming item. Everything else is obsolete. With Chemical Rage and the bonus attack speed, you can wipe out creep waves in no time. With the active, it also punishes enemies for damaging you, which synergizes with Alchemist's ultimate.
Assault Cuirass increases your attack speed even further and gives Alchemist his much needed armor. The armor shred also makes him hit even harder which makes him extremely powerful. This item is also good on a Ganking Alchemist as a luxury item.
Luxury and Situational Items
As there are a ton of items that work in both builds and lot of overlap, I won't go through every single choice. Rather, I'll go through the ones I find most common or most effective.
Skull Basher/Abyssal Blade
Almost always gotten on Carry Alchemist and it's easy to see why. This item is so good it's almost core on him (almost ;>). The low BAT and amount of attack speed items we build make it to where you're almost guaranteed to proc a bash. It's also good on a Ganking Alchemist if you're looking for more raw damage and less utility.
Seen more on Carry Alchemist than Ganker Alchemist. It's the alternative to Blink Dagger as an initiation tool. It gives really good stats like damage and attack speed and the active is both nice for escaping and nice for initiating. It does make you a bit more mana intensive, it's countered by detection, and is more expensive than Blink Dagger. It's also not the best choice if your team is teamfighting as the active is somewhat wasted as an initiation tool. If you ganking as a carry it's a nice item but if you're going ganker, Blink Dagger is generally better.
Boots of Travel
Boots of Travel are a good choice at all stages of the game. It leaves inventory room for more items rather than TP scrolls and is also less expensive in the long run than consistently buying scrolls. They're a great pickup on both a Carry and a Ganker. As a Ganker, you no longer have to walk or spend cash on teleport scrolls in order to gank a lane. You can also teleport right into a teamfight or gank to help secure a kill. As a Carry, it helps make sure you can always be with your team in between farming and fighting. It's also nice to be able to teleport to that nice, big juice creep wave pushing your bottom tower
after you've finish eating your stacks.
Desolator is a fantastic DPS item on a lot of heroes. I'd advise against it on a Carry Alchemist (if you want DPS grab a Daedelus or Abyssal as they synergize much better) but it's really good item on a Ganking Alchemist just so you can stack more negative armor. It's also good for pushing (split push alch with deso+tp+shadow+necro book at the next internationals anyone?).
Mask of Madness
Mask of Madness isn't the worst choice in the world for Alchemist. He has good base STR, he has good regen, and he makes really good use of the attack and movespeed. The main problem is that it doesn't help his farm as much as a quick Mjollnir will and that it makes him too squishy against late game carries. As a Ganker Alchemist, it's likely a better choice if you want to become a secondary sound of right click damage on your team, but most games I'd say avoid it.
Black King Bar and Linken's Sphere
Black King Bar is your item of choice against hard disables, slows, and AOE magic damage. it allows Alchemist to run up to the enemy carry without having to worry about being kited by the support's slows and stuns.
Linken's Sphere is best against big, single target ultimates like Rupture, Doom, Dismember, and Fiend's Grip. It accomplishes the same thing as BKB but it only blocks single target spells. It also gives better stats.
Both items are viable on Carry and Ganker builds. Some lineups may require both but I advise against getting both most games.
Armlet of Mordiggian
Armlet synergizes well with Alchemist to a certain extent. His HP and Regen are great but the item offers very little attack speed. It fits a Ganker/Tank role more than it fits a Carry role, but it's still somewhat viable on both builds.
Hand of Midas
only get if going jew alch build (manta radiance bfury mjol) huehuehuehue
In all seriousness, Midas is an okay pickup if you're getting free farm or you have a good gold advantage from early kills or towers. The main problem is that it delays your Mjollnir, which lets you chain last hit better (which is the point of Greed!) than a Midas. Still, it's an okay item to build to help secure a gold and XP advantage. I'd advise against picking it up on anything other than a Carry Alchemist unless you know what you're doing.
A fantastic luxury pickup for Alch as it helps solve his low HP gain. The regen feels a bit wasted, but the massive 1k HP it gives you is well worth the investment.
Ever since I saw the Alchemist with Heart, 4 Butterflies, and boots able to tank fountain for minutes on end and get consistent kills on the poor, doomed, underfarmed heroes soon to spawn, I wondered if Butterfly has any application outside of just being a troll item. After a few tests, it actually is a decent alternative to Assault Cuirass. The agility gives Alchemist plenty of armor and attack speed and the evasion makes you annoying to fight. Assault Cuirass is overall better in my opinion, but if another hero on your team has built it, Butterfly is a good alternative.
Misc. Caster Items (and other stuff)
The above a handful of caster items that work well on Alchemist. Most of them are more geared towards a Ganker or Support Alchemist, such as Atos, Sheepstick, and Shiva's. Blademail and Skadi are situationally nice on a Carry Alchemist. The stats and HP from Skadi are very good on him and the slow makes sure no one can escape. Blademail is fantastic against glass cannon heroes like Drow, Sniper, Shadow Fiend and against AOE heroes like Venomancer and Phoenix.
Like I said earlier, there are a lot of items that work well on Alchemist depending on your role so don't be afraid to experiment or try something new!