I actually think Kat is fun but yes, she is very dependent upon team compositions, both her and her opponents. The big key is that Katarina does not start fights, she finishes them. Don't rush in thinking you can get a pentakill if you don't have anyone following you. They'll just delete you. Instead, skirt around the sides of the fight, throw out Qs where you can and wait for your opportunity. Once the enemy team has blown their biggest CCs, try to pick off anyone who is low, or position yourself in a relatively safe spot to throw out some damage.
Then once someone is low, the game is on and you basically just roll your combo onto each new target once you get your resets.
Also if your team has a strong initiator or CC, don't use your Shunpo to position in for your ult. If you have a Malphite or Amumu or something, let them initiate and try to walk into range. Start ulting but watch your CC enemies and be prepared to Shunpo to safety. Its better to stop your ult early then wait too long and get caught.