Well you got a few problems that should be fixed. Let me start
First of all,
when you rig the fingers, they should be constructed of 3 bones (excluding the connection to the wrist or other helper joint). When you finish doing this, you should create controls all 3 bones. Starting from the end of the hierarchy (the finger tip joint) you should:
01 - create a tip bend channel attribute (do that through connection editor); then
02 - create a middle bend channel attribute;
03 - for the 3rd joint ascending in the hierarchy you should create a rotation channel attribute horizontal to the hand.
Repeat this step for all fingers, including the thumb. For the thumb you should add 1 extra channel attribute middle lean so you can lean the finger towards the index finger.
Now you should have your fingers set up.
Only put a controller on the tip joints so you can select the hierarchy and then you can start ascending.
02
When you move your bones after binding, they're probably not painted appropriately. Painting skin weights is a real pain, but it has to be done if you really want to see good results. Heat map is the best skinning method, but usually it doesn't work because of an error in your geometry. There is a maya bug with this apparently ;/
Try using Geodisic Voxel or classic linear. Although classic linear is the safest, since it will save you a hell of time with the skin weights.
Start constructing it from the beginning, and you should not have any problems 😀
