I recommend starting boots and health pots, if you are more comfortable going cloth armor or amplifying tome first (with health pots) due to the enemy's mid laner that would probably be wise too. Farming in lane is an important part of your late game strength. Farm with your BOUNCING BLADES or auto attack of course if you can. When auto attacking minions try to farm the ones who are trying to go to the back of your minion wave so your opponent will have to travel enough for you to react to there presence if they are attacking.

For your first back you should be able to purchase Aether wisp. Once you get Aether wisp you now have some roaming power, roam bot frequently for some kills or assists.

Your famous combo BOUNCING BLADES, SHUNPO, SINISTER STEEL, DEATH LOTUS or q,e,w,r is Katarina's bread and butter move.

Katarina is a squishy melee damaging assassin, in other words a prime target for the enemy team. They will want to eliminate you as fast as they possibly can. you need to wait until SOMEONE ON YOUR TEAM HAS ALREADY ENGAGED. Make sure their tank is not with the ADC or is attacking someone else. Then finally its your time to shine, jump in with a BOUNCING BLADES then SHUNPO immediately after shupoing then attack with your SINISTER STEEL. Following up with a DEATH LOTUS remember death lotus (your ultimate) can only target 3 enemy champions. This may sound really long and slow but once you use it in action its a different story.