Rock climbing is largely a skill sport, so to get better, climb more. Being generally fit also helps, but to be a better rock climber, you need to climb a lot.
If you want to specifically do extra work for rock climbing you should start off with identifying your weaknesses. From my experience in my climbing gym, the biggest weaknesses are grip strength and flexibility.
Things you can do at home
You can train your grip with something like the Black Diamond forearm trainer. You could probably just find a block of wood and just squeeze it. Real holds don't have any give after all. Don't use a regular grip trainer. Your closed grip is probably relatively strong. You probably want to train your open handed grip positions (like you're holding a fat hold).
You can also try Metolius Rock Rings. They are like a pair of mini hang boards that you can just hang somewhere. They come with an example program.
Flexibility is also worth working on as increased flexibility open up moves that you may not be able to perform at the moment. I find leg flexibility more limited than shoulder/arm flexibility. I'm not confident in prescribing stretches but perhaps someone at your gym can give you a few pointers.
Add drills to your regular climbing time
Use the campus boards and hang boards in the climbing gym. You can do various drills before you climb:
- Just hanging
- Hanging in the middle position of a wide grip chin up (so that your arms are beside you and your forearm and bicep form a right-angle)
- L-sit hangs
- Pull ups
- Off set pull ups (one hand lower than the other)
- Various patterns to get up the campus board
Ask someone at your gym for advice and to show you drills.
Otherwise maintain good general fitness.
Edited to try and answer the question more directly. Thought the Edit Summary field would add an editing note. Does not seem to be the case. Edited again to put edit note