Please refrain from picking Axe. They glitched Battle Hunger in the latest patch and you can't get rid of it by killing a unit. Please don't be a ****; don't play Axe until they fix it. Thank you.
This build's main focus is to help your sidelane getting kills as much as possible. It's not the easiest way to play Axe I agree, though I find it to be the most efficient one if you know whoever's playing the long lane is decent.
Your main skill will be battle hunger. It is way overlooked by most Axe players as most people would rather spend points in Counter Helix. Battle Hunger is an outstanding DPS ability: It slows, it buffs your speed(Stacks if you BH multiple targets) and most importantly; you can't deny someone who's under Battle Hunger.
Battle Hunger will help you catch your target for an easy Berserker Call or to catch someone with your ulti.
You will have a harder time jungling since you're not leveling Counter Helix past level 1 until you're level 12. This is the reason why you need to know you can trust your lanemate when you start ganking as your ganks will determine whether your mid game will be smooth or not. Don't worry though, you are well able to jungle with a level 1 Counter Helix :).
Jungling with Axe is pretty straight forward. The most important thing in order to get a headstart is to start by stacking the easy camp at 0:53. Make sure you haven't spent your skillpoint on Counter Helix yet because you won't be able to stack the camp if the starter mobs are the ones who can heal.
After clearing the easy camp, you will want to go stack the pull camp(If not warded) and then pull at 2:15.
IF the pulling camp is warded, go sit in lane a bit until you can stack the easy camp at 2:53. DO NOT try to take on the big camps if you're up against a warded pull camp unless one of them has the group of 4 weak satyrs(We love these :3).
By then you are level 3. This is when you will start using battle hunger on the weakest/support hero you're laning against. When you battle hunger an oponent, you have two choices: Go for the kill or go back to your farm. You're unlikely to be able to get a kill with your first battle hunger, but if your lanemate has harassed nicely, you might be able to charge in for the kill. You have two shields thus you can tank a creep wave like it's nothing.
After this first engagement, you must chose which item you'll be starting with.
If you're confident in your farming ability and have full hp/still have your potion; go for your arcane boots.
If you had some problems and need the hp regen; go for your Ring of Health.
If you went for arcane boots, you will have less farm from jungling although you will harass the lane and might get a kill or two.
If you went for RoH, you won't have the mana to battle hunger as much as we'd like to but you will farm the jungle with ease.
When you have both Arcane Boots and Ring of Health, simply keep jungling while poking from time to time to battle hunger.
This is pretty much it for jungling. Remember to stack the easy camp as often as you can.
The mid game transition is usualy when you'll start getting a lot of kills/assists as you have both survivability(With Vanguard) and mana. Battle Hunger's movement speed buff will allow you to catch up to your opponent before having blink.
During the mid game transition, you will want to keep on jungling & keep on ganking your sidelane. You can start ganking other lanes when you get your blink.
Mid game is the nicest part of the game for Axe. Once you get blink, you can start moving around the map and gank every lane. Make sure you carry TPs to help out allies who get ganked.
Your next item is Aghanim Scepter. Aghanim scepter gives you both health & mana while changing your ult's cooldown to 6 seconds... You may not have a lot of gold and your aghanim could be delayed. If that's the case, start building it with the point booster and finish it as soon as you can. Scepter really shines during team fights ever since Axe's ult got buffed, adding movement speed to allies after you get a kill with it.
Keep ganking, keep stacking & jungling... Take part in team fights.
After Aghanim, your next stop is Blade Mail.
Late game is the time when carries get fat and dish out a ton of damage in no time. Battle Hunger is amazing, but it loses effectiveness the more your game drags out. This is why you need a Blade Mail(It's core on any Axe guide anyway... Or should be)
No need to explain how Berserker's call & Blade Mail can be deadly against any fat carry :).
After Blade Mail, the game should be pretty much over. If not, you can start making a Sheep Stick or an Assault Cuirass depending on what your team needs. Heart of Tarrasque could also be a possibility(You can swap your vanguard for a Heart).
That's all! I hope this will help you play Axe in an unusual(Yet deadly) and fun way :).
Keep on chopping heads off!