NOTE: This is my first guide, so I don't really know how to set this out properly yet, please be patient :)
The build for this guide is fairly common in Axe players and includes Vanguard, Tranquils, Blademail, Blink Dagger, Aghanim's and Heart of Tarrasque/Battlefury.
Vanguard: Bread and Butter item for Axe, great for keeping him alive and in lane where you can deal out the hurt, get this item FIRST. Note: You will need to get to a secret shop at some point, easy if your Dire Bot or Radiant Top but difficult if your the opposite so be wary of ganks.
Tranquil Boots: Some people frown at these boots because they prefer Phase for the movement speed boost (which to their credit, Axe does lack) but Tranquils help in the early stages of the game. They can bought entirely from the side shop and the heal will help when you've took a bigger ***-kicking than you hoped for. Get these SECOND.
Blink Dagger: Again a must-have for Axe, hes pretty slow even with boots so the Blink will help with escapes just as much with setting up attacks. This works well in a combo will Berserker's Call, Counter Helix and Blademail. If your farming/killing well, get this THIRD, however if your having a spot of trouble getting the gold get this FOURTH it should not be picked up any later than that. Note: Blink Dagger cannot be built, it comes complete but is pretty expensive so be careful about losing gold to deaths.
Blademail: Blademail will rape if you time it right with Berserker's Call. Pop it as soon as you Blink in and use BC, the enemy will be forced to kill themselves, especially if your lucky with Counter Helix spins aswell. Blademail will also help you escape if you get ganked, however still be wary of stuns/traps/blocks etc.Get this item FOURTH or THIRD if your having trouble getting the Blink but can pick up parts for Blademail.
Aghanim's Scepter: Aghanim's will give you a decent stat boost aswell as lower the cooldown of Culling Blade, meaning you can chop heads on a more frequent basis. This isn't a must have item for Axe but it helps nonetheless. Buy this at any point after the first four items.
Heart of Tarrasque: Heart of Tarrasque is an excellent item for Axe as it will give you health, regen and damage. The reason why it isn't one of the first items to get is because its damn expensive, with the Reaver costing 3200 flat, so building it can be difficult if you aren't getting kills or are getting killed alot. If you are doing very well, get this as soon as you can, bypassing all the other items if necessary, if not get it as soon as possible.
Battlefury: Another item I was flamed for using one time. Battlefury is one of my favourite items for melee heroes because of the splash damage effect as well as the decent damage boost, unfortunately as Axe doesn't attack that fast you don't really put out enough DPS for it to be viable. However if you've managed to assemble Blademail, Heart of Tarrasque and Blink Dagger you can create a deadly whirlwind of damage despite your low attack speed. Get his item in replacement of another that you don't/need, maybe Vanguard or Aghanim's Scepter.
You want to max Battle Hunger and Counter Helix ASAP as they are your biggest damage tool early on. Grab a point in Berserker's Call to get the enemies attention if you think you can grab a kill or protect your teammate. Get Culling Blade whenever it is available.
Battle Hunger:Throw Battle Hunger onto the softer of your two enemies, do it often but don't spam it too hard else you'll burn a lot of mana. If you have a ranged lane partner, it would be best if they were to harass the enemy to keep them away from killing creeps and removing the Hunger. Even if your lane partner is a carry it will better benefit you both. Battle Hunger becomes less effective late game but is good for picking off weak enemies who escape a team fight. This will pick you up a lot of kills early if used correctly.
Counter Helix: As you have no control of Counter Helix you don't really need to worry about it. You do want it maxes quickly though so it does the most damage. If you get lucky with the cast rate (which is only about 17%) and a good Berserker's Call this can f*ck enemies up majorly. The nifty thing about Counter Helix is that cast even if your running away, so if your being slowed by a ranger while being hacked by a melee, the ranger can trigger your Helix and damage their own team mate.
Berserker's Call: This will attract your enemies to you and cause them to attack you. It gives you a 30 armour boost which will help survive but you need to be careful, it can go really well or horrifically bad. Combine this with Blink Dagger,Blademail and pray for Counter Helix to proc and you can shred down multiple enemies in very little time. Use it whenever you feel the need to attract the creep wave or the enemy team, normally when setting up kills or to protect the softer heroes in your team. Be wary however that the range is fairly small so you have to get close to do it which can be dangerous if you don't have Blink Dagger.
Culling Blade: Without a doubt one of the best skills in the game simply because its so damn fun. Culling Blade will deal a relatively small amount of damage in comparison to other Ultimates however if the enemy is under the right kill threshold it will kill them instantly giving an awesome animation of you slamming them into a bloody mess. This will not activate if you get the enemy under the threshold, example: If the enemy has 350hp and the threshold is 300hp, the damage the attack actually does will not trigger the kill it will just take the damage off. They need to be ALREADY UNDER the threshold for it to work.
The skill will tell you in game what the thresholds are, I would tell you but I've forgotten and I'm too tired to look it up.
In the early stages of the game focus on getting decent farm and harassing the enemy heroes if they get too close. Pop Battle Hunger to weaken the softer targets and let Counter Helix work for you. Don't go balls deep too soon, work on getting a Vanguard and Tranquils first. However if the enemy slips up and you can kick the **** out of them go ahead :)
Work towards getting a Blink Dagger and Blademail, by this point you should be a decent enough level to take a fair beating without it being too dangerous. When team fights kick off, don't be shy you are a tank so go tank up. Use Berserker's/Blademail/Blink if you can and don't forget to keep an eye out for easy Culling Blades, your team mates may cry "Kill Steal!" but a fed Axe is a happy Axe and they'll thank you for it in the long run
By this point you should have a decent build and hopefully kills under your belt. You want to stick with your team still as you will be needed in team fights but generally its just fight/kill/leave, rinse and repeat and you should be chopping your way to victory in no time :D
Good Day Sirs! (and Ladies)
Pros and Cons
Axe loves to chop squishy targets, seek them out in team fights and you'll kill them quickly.
Axe is hard as nails, you can take a beating as well as give one out.
Axe isn't skill intensive, you don't need to spam your skills, just use them carefully and effectively.
Culling Blade just owns. It does. Fact.
Axe is a melee hero and will suffer against range heavy enemy teams. However if you can close the gap you'll tear them to shreds.
Axe isn't the fastest of heroes so before you get your Blink you'll be relying on your boots and skill to escape from ganks/failed offenses.
The item build is pretty expensive so you need to be careful of gold loss as well as gold farming.
If none of your team mates has a disable/stun/snare etc you could really suffer in a fight if a Berserker's Call attracts the wrong kind of attention.