League of Legends (LoL) Question: how should you build evelynn?

I would go AD Tanky, AP not only falls off, it's also more unreliable. Tanky you have much more of an impact, since you will be so hard to kill with your shield + some tanky items, while still dealing damage. If you go in to land an ultimate while stealthed, you have a significantly greater chance of surviving as a tanky bruiser than as a squishy AP Mage. Both work though, and it's smart to try out both, so you can test it out for yourself, you learn more that way.

Runeglaive is really strong on her. I recommend buying Stalker's Blade - Runeglaive, Sorcerer's Shoes, and Rylai's Crystal Scepter. If you want more AP, go for items with some defense like Liandry's Torment, Zhonya's Hourglass or Abyssal Mask. You can also go for heavier defensive items like Frozen Heart, Randuin's Omen or Locket of the Iron Solari.

I was a fan of the Warrior-into-tank-items build a while ago but I think Runeglaive is a lot stronger currently. It helps your sustain and clear speed a lot once you complete it and Evelynn is great with Sheen procs.

One of my favorite parts about eve is that you can really build her however you want and however suits your playstyle. Since both AD and AP builds are viable, its up to you whether you want to build her burst AP or DPS AD. Right now a lot of the pros are building her ap/mp but mess around with her and find what suits your playstyle