Yes, the text you provided is grammatically correct. However, there are a few minor improvements you could make for clarity and flow. Here’s a revised version:
"My character is named Jimmy, and he loves to go outside in his backyard. Jimmy is usually found engaging in activities like jumping or running there. When he’s not in his backyard, he has adventures throughout the world, which involve defeating monsters for fun whenever he has nothing better to do. After his adventures, he returns home and sleeps."
These changes provide a bit more clarity and conciseness, but your original version is still well-written!