How often should you climb per week?
There is no definitive amount of times someone can climb per week. It just depends on each person, as each person has their own fitness level and athletic history.
When I started climbing, I climbed around 4 times per week, each session was 4-5 hours long. Needless to say, I injured myself. I toned it down a lot to just 2-3 times per week at most, and also took some time off here and there to give my injuries and muscles more time to recover.
After this, I create a better schedule of climbing 2-3 times per week, for around 2-3 hours each time. I came from a background of weight training for quite a few years. This did help with my climbing as I was able to pretty quickly progress to v6, but, I did overdo it in the beginning. With my new schedule, I was injury free and haven't been injured since.
So even if you have background in something, or you are generally fit, you can still very easily overdo it. Climbing is different from most sports, it uses your muscles differently, so it is just very easy to overdo it.
The best way to know how often you should climb per week is to just listen to your body, pay attention to it, and don't ignore the signs. If you do, the next time you get another sign may be too late.