EDIT: That game was mid vs a Heimer.
I've been successful carrying in mid, top, and jungle. Build is always rush nashors tooth, (If you have enough for stinger get stinger, if not get fiendish codex), after completing Nashor's you'll notice an immediate powerspike and this is where things get fun. You don't neccesarily need any kills before you get your Nashor's completed, but you do need to work extremely hard to not die and farm. After nashors, I grab sorc boots, then deathcap, then void staff.
One important thing to note about the Nashor's rush is, it gives 20% CDR. Throw a blue pot on top of this and you have 30% without masteries (I use attack speed, not CDR in offense). Add blue buff and you've got 40% CDR extremely early, without sacrificing runes or masteries, and this is where you can shine.
My 5th/6th items nearly always vary game to game, but those first 4 are pretty consistent. The one exception would be rushing a Morellicon (AFTER NASHORS + BOOTS2) against someone with heavy sustain like warwick.
Jungle Azir is very rough starting out, but he clears so fast once you complete your jungle item (which I've only done the AP version so far, but I'm highly contemplating feral flare Azir).
