Ok! Katarina and diana have both a similar assassin type of play-style, but there are some adjustments to make. When diana goes in, she has a harder time to get out then katarina. The higher elo you will be and the harder you will get punished for a bad all in decision, this should develop itself with experience. Also, when the enemy team is stacked together and your team is close, you might want to actually flash in 1v5 since you excel in aoe damage: you will get them so low and closer with your e that your team should be able to clean up. Mechanically, you always want to keep track of your passive since its the strongest damage source of your kit. If your passive is ready, any all in is way stronger and way more justified.
Finally, after roaming and getting kills, you often want to stay and get a tower. Everyone says it, but ill say it again, she excels at taking towers: use this strength.
Also since you are a katarina player i assume you play diana with ignite and this is totally fine. Diana is good at split-pushing and she still does it with ignite, but i couldnt really explain to you how to do it since im a tp diana player.
Oh and diana is super strong at ending games since she is a very high threat beside objectives where you stack your passive. Basically, if your team is sieging, you can zone people out of defending towers or simply jump on anyone close enough, kill him, rinse and repeat.