Axe is a very strong initiator and disabler. He has an aoe disable capable of piercing spell immunity. He can easily take down lots of heros with just few items, even if they are higher networth than him.
Axe can easily cut waves by farming behind enemy towers with his counter helix. After clearing the wave instead of staying idle, you can farm jungle or stack it for later use but be careful not to lose the stack. You can also drag the creep wave towards the neutrals and farm them together.
Battle hunger can be used to harass enemy and stack multiple camps if timed properly.
Time your battle hunger with enemy last hits to maximize its chance of staying for longer duration like when enemy is trying to last hit under tower or when all of your creeps focus one creep and that creep dies, it will take few seconds before the next creep is close to death.
Battle hunger also affects creeps, you can use it to gain a slight movespeed boost when needed like you can cast it on the neutral creeps when chasing a enemy at night time.
Remember that Axe gets the bonus armor using Berserker's Call which can help you to survive longer in some situations or sometimes escape. When enemy is taunted by the call, you canwalk or use items like fore staff to force the enemy to walk towards you and compromise their position.
With vanguard, 2-3 levels in call and 3-4 levels in his counter helix, Axe can clear 2-3 ancient stacks depending on their type.
This applies to most initiators/gankers, stay hidden on the map. Just by staying hidden the enemy is forced to play defensively and lose out on farm. But don't always stay hidden as you might lose out on farm or towers.
Counter helix is pure damage which means its damage won't get reduced or get blocked by spell immunity and also it deals damage even when he is disabled.
Aeon disk counters axe hard. So have some dispells on your team to help you with that. Also the enemy team cause use force staff or euls to help their allies when they are under Berserker's Call. So consider getting bkb or lotus orb.
When below the kill health threshold, culling blade will instantly kill the enemy unit even when its under effects like borrowed time or false promise. It can also be used to kill a courier. Cooldown is reset only when enemy hero is killed and not any other unit.
Aghanim's Scepter is rare situational item but it is powerful. As of patch 7.19 most cores get bkb to have impact early on. When game goes to later stage bkb duration gets low. Battle hunger has a cooldown of 5 seconds and mana cost of 80 at max level. It can be easily spammed, forcing the enemy to get more defensive items or get their damage output lowered by 30%.
Using the console or config files, we can bind "toggle" and "dota_health_per_vertical_marker" to an unused key and use it to change marker on HP bars depending on the level of Culling Blade, like key "k" can be bound with "toggle dota_health_per_vertical_marker 250 325 400" which will cause the 'k' key to cycle through the listed values.