I've picked up Katarina a couple days ago and I don't seem to do so well with her as I did with Evelynn or LeBlanc.
I build:
>Rabadon
>Deathfire
>Hextech Gunblade
>Abyssal Scepter
>GA for tankiness
>Adlib (I always leave an ad-lib slot in my builds for whatever seems appropriate)

Can anybody give me some tips for Katarina? I.E. What's the best way to prevent them from walking away from your ult?

Oh, and would a duo lane Katarina/Lux be viable? Kat as APC and Lux as Support.

well kat does well with certain champs, if you have a jax on your team and you see him getting ready to stun their team or even if its not a team fight, that one person. thats a chance to ult, while they're cc'd and cant get away. another thing is to watch their team, know who can and cant stop your ult and try to wait out ultill their cc is used before you go in and try to use your ult. amumu's ult is a nice way to get off a decent kat ult, and so on. if you pick kat you really want to make sure their whole team cant stop or ult or you better have a team that you can rely on to cc so you can ult.

some people hate it, but i use rylais on kat to prevent people from escaping, its also good for teamfights, because people cant rush up to your adc. its even more important if you know that you wont be able to carry the game as kat, because of too many disables.

its all about dat_counter_pick

My #1 suggestion for Katarina: Smartcast.

I do that already. Thanks for the suggestion anyways!

Sunfirecape+warmogs and any other items after that gg

Not sure about warmog's, but sunfire isn't actually that bad an idea. Helps you land those resets from the assists with the AOE damage, and gives you some tankiness.