I played AP evelynn for a long time, but have found a better dps/atk build for her that seems to be overall better at killing in team situations.
ability tome + health pot first
Get sheen next -> first item that almost doubles your damage for a total of under 1300g (It will proc it seems like every other hate spike)
Next get some regular boots
Then a ruby -> Phage
and Boom, Trinity force
upgrade to boots of mobility so you can chase and kill that person with the elixer
And if they don't already hate you whilst only having 2 items, go get a BF sword, then Pickaxe, then cloak and upgrade to Infinity Edge.
I've only ever had 1 game that lasted past this point, so I bought a last whisper just to top it off.
They die so fast and can't run away. Skill build with 1 pt in hate spike, shadow walk, ravage, ravage, ravage, ult, ravage, shadow walk, ravage, walk, walk, ult, walk, walk, spike, spike, ult, spike, spike.
I only use spike to proc trinity force and for early game creep kill steals.
Thats how I roll with eve.