Read her ability tooltips carefully.
They all do MAGIC dmg therefore MR counters her, so she needs MPen to increase her output.
However BB scales with AD, shunpo scales with AP and the ult scales with either (picks the highest).
The type of damage an ability does is separate from what it scales with. On most champs AP=magic, AD=phys but it isnt always the case.
Kats not really the best champ to be playing atm anyway. She is mostly about her ult and it is too easy to shut down. However if you really wanna play her the best route is after you get the sorc boots and first BF, grab a chain vest and a negatron cloak (be smart - if they're all phys drop the cloak). This will give you some staying power when you jump in. As the game goes on, keep grabbing BFs as needed and upgrade to Randuins and FoN when the enemy is packing damage.
Always run with flash/ignite. Flash gives allows you to escape or initiate (then escape with shunpo). Or even better you can pull a slingshot around a team fight by flashing to the side and then shunpo to the enemy squishies in the back. Ignite helps with early kills, which are essential to start snowballing.
Most importantly: wait. Patience is your best friend. Chuck BBs until the fight starts and then wait until everyone is half dead and the CC is used. Then jump in, ult, get 3 kills, ult again, win. Wont always go like that but if you go in early and get killed/CC'ed it'll never happen.