Not Updated For Current Season
This is how I play Katarina after the patch.
I use Q to harrass safely while I Rush a Rylai's for survivability and an amazing slow that stacks with gunblade's slow active, while gaining awesome scaling from gunblade stats.
Summoner spells:Ghost/Flash and Ignite
NOTE: YOU NEED A SOLO LANE
For early levels apply pressure, spam your E on the foe and auto attack to follow up. Not many champions can out damage you with this level 1 harrass. Kite you foe while you wait on cooldowns.
After you level to 2, start using the E/Q combo, followed by an auto attack. This is where you begin to dominate the lane. By level 3 they should be popping health potions where you use W/Q/E Combo for the rest of lane phase until level 6 where you can either kill your lane enemy, or if they are forced back to the turret GANK A SIDE LANE with your ultimate. This thing will get you a kill every time it is up, so ganking a sidelane can score you a double even.
Mid game is a bit different, you cannot play as aggressive with E, but this is why gunblade kat shines, your Q. You spam the **** out of Q with W when you can, The damage is really noticeable on the squishiest target, but IT IS SIGNIFICANT. Proceed to poke on cooldowns until you are ready to E/R/Ignite them down.
Lategame: Play smart and do not go in first. You have an aoe ultimate but that doesn't mean you are going to kill everything. Let someone engage and go in shortly after, do not E in. Save E for after they run from your ultimate. Instead, run/ghost in, gunblade active the squishy or carry and W/Q/R/Ignite him. This should grant a kill as they are extremely slowed and bursted down fast.
WHY GUNBLADES OVER BLOODTHIRSTERS: Your Q and R both scale with ap+ad. This means you want to buy items with great stats on both as well as having other unique utility. Since Kat is melee ranged a lot of the time her spell vamp keeps her alive very nicely and even if you spam and harrass with a few Q's the spell vamp is nice. With gunblades you get nice auto attack damage, nice lane survivability, an awesome slow/burst active and great scaling on your abilities.