There are actually different types of grip strength, the type you use when balancing on your hands is going to be different to the type you use pulling a heavy deadlift. Yes, there will be some carry over, but since you're trying to accomplish a free standing hand stand, specificity comes into play.
First off, I would recommend stretching out your forearms regularly using a kneeling forearm stretch (on your hands and knees, rotate your arms outwards until your elbows are facing forward, palms flat, and lean back gently to stretch the forearms). When you do this, make it a more active stretch by gently resisting the stretch.
Although it sounds like an odd suggestion, stretching like this can actually help condition the forearms to having to apply force in this manner (very similar to when you have your palms flat on the floor in a handstand position).
Secondly, you can train the Crow Pose as a progression into a free standing handstand. To make it a little more interesting when you get reasonably good at it, gently apply pressure through your hands to rock yourself backwards and forwards.
The reason I stress gently so much is that to start with, your forearms may quite tight and you don't want to risk an injury (I've seen a climber trying to get rid of a forearm pump badly pull something by trying to stretch too hard).
Don't forget to stretch your wrist in the other direction as well, since general wrist mobility and strength is important for a decent flat handed free standing hand stand.
Lastly, just a general handstand tip, spread your fingers as much as possible; it'll help with balance and control.