i think many people were misled by the champion spotlight.
when i first play Elise i also thought she's tanky ap like Diana, but i was wrong.
The reason why you shouldnt build her ap, is because her unique ap ratio. her Q only gain 2% for 100 ap, and the base damage does't add up, and 2% of current hp or missing hp is really not that much.
her W in human form is also kinda useless, it get easily blocked by minion.
Since you can't get more burst by building ap like Diana, you need to find another way to play her, which is included in my guide- atkspeed tanky Elise.

Q(human&spider) - although her Q doesnt have good ap ratio, but it still does some good dmg early game 260+8% of current life at lv5, thats why you should max this first.

W-(human) - nothing is outstanding for this skill, the only good thing is that your enemy can't run away from it if he's alone.
W-(spider) - gives you attack speed which is the core of this build. the life steal is also helpful.

E - (human) - really good cc skill, lvling it up only decrease cd.
E - (spider) - this is the skill to determine you are a good spider or not, lvling it up only decrease cd so lv it last.

R- MORE SPIDERS!

when i first play Elise, i build her like Diana, but building ap really didnt give me any dmg plus i died really fast. when i found out it's due to her unique ap ratio, i decided to try another build, which is bulding attack speed and more tanky.

the concept of this build is to bring out her bonus magic dmg on attack and to become more beefy inorder to do more dmg.

in early game, you can always harass your enemy with your q and use your ranged attack, if your opponent decided to have a fight just makesure you used qw in human form(save stun) then switch to spider form to fight him, most likely you will end up wining and then chase them with E.
if you lose just switch back to human form walk back stun and kite them. So far i have supressed jax jayce and irelia on top lane using this method.

Elise is really strong on top lane due to her Q and spiders, after you grabbed the wooden hammer, you can go around to gank or stay on top lane and keep supressing your opponent.

any questions just leave comments, my english is not that great but i will try my best to explain.