Strengthening the weak hand grip is a good idea and the suggestions offered by posters so far are good ones.
Train your weak hand the way you did your strong one. Fundamentals first, sight picture, trigger squeeze, two-handed weak hand shooting and finally one handed drills, all low speed for accuracy. You may have master eye issues and these can be addressed through body mechanics. As your skill begins to grow, increase your speed without sacrificing accuracy. Then move into the same type of combat drills you perform strong handed.
Learn to safely draw, manipulate, clear malfunctions, and reload your gun with only one hand or the other. This should only be done with an unloaded gun and no live ammo present. (These are essentially non shooting drills and there is absolutely nothing to be gained by doing them with a hot gun.) Pretty important stuff to know if things go bad.
Once you begin to acquire your basic skills, work them every time you go to the range to build and improve. I sometimes take a day and spend almost the whole session on my weak hand. I try to make each shot I send downrange contribute something to improve my skills.
Respectfully,
DarkSoldier