Nothing wrong with AP, I just found I often do better with the AD build. Evelynn plays different from most assassins in that she has no heavy burst. Most of her abilities scale off of Both AD and AP... BUT she'll still need to land a few AA's to finish the job; Add in the fact that ravage activates on hit effects and grants turbo attack speed makes the life steal grant her a lot of trading power!
Still, she's kinda squishy for a melee anything... No mater how you build her you have to pick your engages carefully! A carless all-in will likely get you killed. ...In this regard, she's pretty much the most REALISTIC Assassin in the game! (IMO :P )
As for Dead man's and Ice born, It's not just Eve, I usually build at least one def item on every champ I play. (
on
is gross!)
Lich Bane isn't the worst way to go, but I prefer Ice born for it's on hit effect. Eve's ONLY CC is her ult which has a huge CD. I've secured a lot of kills with the perma-slow this grants. Between it and Eve's W, VERY few champs can escape w/o flashing!
Evelynn is one of those champions that looks simple on paper, but actually requires a decent amount of practice to play properly. It took me a long time to get her right! If you want to go AP, Maybe consider
and
.