R>Q>E>W with one point in each by level 3.
21/9/0 As much AP as possible, health/armor/MR in defense.
Ignite/Flash. Pretty much the only two you should run, anything else is sub-optimal for combo-killing and escaping on Kat, which is all she can do really.
If owning hard: Gunblade -> Rylai's -> Rabadon's
If struggling: Rylai's -> Gunblade -> Rabadon's
If getting completely eaten alive mid: Abyssal ->Rylai's-Rabadon's
After that, possibly Void Staff, GA, Rabadon's
Kat is a mid lane champ that is almost entirely dependent on match-up. Enemies with skill shots like Xerath, Brand, and Ahri can be defeated, but it's pivotal that you actually dodge their skills. Enemies like Le Blanc, Kassadin, Cassiopeia, Mordekaiser, Ryze, and Veigar will stomp you most of the time.
Kat is only as good as her ult, and any hard cc will almost completely cancel out your damage. This means you either don't play Kat against these sorts of champs, or be extremely patient and wait for them to use their cc. In the case of Mord and Ryze, they're too tanky/healy to really kill while still having large damage output.
Farm up until 6, harass with E+Q if possible, and go in for the combo. Ignite->W->E->R->Q and if they're half health or less they should be dead, unless of course you're cced during R.
Kat is kill dependent, and can snowball hard with enough gold and kills. Shut her down though, and she's negligible for most of the rest of the game.
In team fights, use Q from afar until tank engages. It's important to keep track of your enemies' cc during the fight, only go all in once you're sure most of it has been used. A well timed ult from Kat can be devastating to the enemy squishies, which you should be targeting, and if it lands you a kill or assist will reset your cooldowns. Be ready to use your cooldowns when this happens, it's a waste to not follow up.
Other than that, practice makes perfect. Look for opportunities to shunpo away using your team mates or wards. Much like Lee Sin, you can carry wards with you to place over walls and shunpo to, allowing you to escape.