I have not worked on a J-frame rimfire, but I have worked on J-frame centerfires and on K-frame rimfires. There is some good advice in this thread, but I have some further thoughts on working on the model 63.
I believe the factory stock mainspring for J-frame rimfires is a 12lb spring. For centerfires, the stock mainspring is 8.5lb. The heavy mainspring is required because rimfire rounds really do require a hard hit. But the very heavy mainspring insures that the double action pull will be tough. Some people have reported good results using a Wolfe 9lb mainspring in J-frame rimfires. The 9lb spring is listed as an extra strength spring for centerfire J-frames. However, other people have reported that the 9lb spring was not fully reliable in rimfires.
You can also reduce the weight of the rebound spring, but there are a couple of cautions about this spring. First, the rebound spring needs to be balanced against the mainspring. If the rebound spring is too light, the trigger return is sluggish and the overall trigger pull just feels wrong. A brisk trigger return is critical if you want to shoot rapid double action.
Also, using a lighter rebound spring can make the single action trigger pull too light. This can make the single action trigger unsafe, and may also cause the single action engagement points to wear improperly. I consider a 2.5lb single action pull to be the absolute minimum, and some people say the minimum should really be 3lb.
Of course, it is necessary to have a trigger scale to accurately measure the single action pull. And I recommend getting a trigger pull scale that can accurately measure the double action pull before starting any project like this. Without a scale it is very difficult to judge the effect of each change.
Also, I personally do not change springs unless I am also prepared to polish the internals. But polishing the internals of a revolver requires knowledge and practice. Jerry Miculeks's "Trigger Job" video is a great introduction to working on revolvers, and I highly recommend this vido for anyone who wants to work on Smith & Wesson revolvers.