Is it possible to transfer my GTA V online character from my PlayStation 3 to my Xbox 360? I don't want to start all over.
According to the information from Rockstar Support found here, You can transfer your progress to next generation consoles(Xbox one, PS4) and to a PC with the help of your Rockstar social club account. They have not mentioned about character transfer from PS3 to Xbox 360 in the provided link.
You can give it a try if you want with the procedure mentioned in their support link which I have given but, you will have only one chance and there are lot of limitations. Proceed at your own risk.
You can also post a question in Rockstar support for a definitive answer for your question.