Axe: The 1v5

Axe is not a carry. And Blink Dagger is not a situational item, it is the main item you should rush right after your Boots of Speed and Ring of Health. Axe without Blink Dagger can't do ****.

NEVER go Tranquil Boots + Vanguard: the latter already gives you a ton of health regen, you should pick something like Phase Boots instead of overlooking the useful actives of the other boots. Or, if you want to keep the Tranquils, you should opt for a quick Blink Dagger (maybe a Vitality Booster before it, which will be turned into a Heart of Tarrasque later) or even a Soul Ring, just to abuse the famous Tranquil + Soul Ring combo.

Aside from that, I don't see anything wrong: in pubs, where Axe gets easily fed thanks to Battle Hunger on low-hp players without TP scrolls, you should get plenty of kills and easily build DPS items.

About the skill build, I prefer to leave Counter Helix at 2 until level 9 and max out Battle Hunger with one point in Berserker's Call, but it's personal preference I think.