If you haven't done so already, check out /r/KatarinaMains, they'll help you out. Also, if you want, message CS_Emerald (Katastrophical) and he can answer most of your questions.
Anyway.....
General tips:
*Shunpo is an auto reset
*E->W->AA->wait for enemy to walk back->Q->E(AA) is your general laning combo if you're going for the all-in
*E->W->Q->R->E(AA) is your general combo, switch the R and E casts depending on the situation (R first for burst damage, E first for chase potential)
*Daggers have a large(ish) range if you cast Shunpo right at the edge of their hitbox
*Shunpo has a cast time, meaning you can "dodge" CC similar to Tristana's W
*Gunblade is your major powerspike....and I do mean major....
*Always E to your opponent in lane if they step on your dagger since if you E to your dagger your E does no damage to them
*Vs opponents with escapes, delay your Q cast until you think they're going to use it. This way the Q will follow them through their escape.
*NEVER USE BILGEWATER (bad skin lul)
Early game:
Learn about wave manipulation if you can. The first wave+first melee creep = level 2, so if your opponent isn't clearing faster than you you can Q the first melee creep of the second wave, kill it with an auto and then E to your Q dagger if your opponent steps too close.
Level 3 is your main powerspike in laning, you can kill basically any mid laner provided they burn cooldowns and/or don't have a large minion wave.
Start with either Dark seal, Long sword or Doran's shield. Dark seal for sustain, Long sword for cheese and Doran's shield vs AA based opponents (e.g. Orianna, Yasuo, Zed, Lucian, Talon). However your starting item is up to personal experience as well, e.g. I take Dark seal vs Zed.
Build double long sword, or hextech revolver first. If you have enough gold for revolver get revolver (most of the time), otherwise get double long sword into revolver. Build Sorc. boots following this if you want to roam and/or your lane opponent is weak enough for you to delay Gunblade.
Mid game:
You probably have Gunblade by now, abuse it by either killing your lane opponent outright or roaming (generally bot lane) to help your team out. You rule skirmishes, and depending how fed you are you can either solo 2-5 enemies or wait for your team to engage and then go wild
This is Katarina's strongest time in the game, so make the most of it.
Late game:
Stay with your team and go in when major cooldowns are blown and you're in a position to hit your daggers. That's all I have to say. Late game is pretty boring for Kat.
How to avoid dying:
*Ward up
*Purchase a control ward every back if you have enough gold (exceptions can apply)
*Pick your fights carefully
*Be consistent with your combos
*Know how each champion works on both your team and the opposing team
*Don't overextend for kills unless you're confident in your ability to play Kat
*Don't fight Vayne if you're not fed ;^)
