So for example when I move an arm of my character and his hand moves relatively with him, everything is fine. When i want to change or turn the fingers though, the axis play against me. This is what I mean:
Help is much appreciated.
It's difficult to accurately determine the meaning of your question, but the video helps. I think you mean that the mesh that looks like a link-knob -- near one end of the bone -- seems to distort when the bone is rotated?
What may be necessary is for you to adjust the weighting of the vertices nearest the bone pivot end -- perhaps to the value of zero.
There are a couple of ways to do this...
METHOD 1: Weight Paint tool:
When you parented with Automatic Weights, blender would have created Vertex Groups that match the names of your bones. Select the mesh and change from the Object to Weight Paint mode, and you can paint a zero Weight (dark blue) onto the vertices you want to stabilize:
METHOD 2: select and adjust weights:
With the mesh in Edit mode, select a single vertex around the problem area. You'll see the bone weights listed in table form:
Adjust downward (to zero?) this vertex you want to stabilize. You can then use the Select box or Select circle or ctl-+ to add nearby vertices to the selected and use the Paste Weight to Selected (vertices) to adjust the whole group:
I don't show but you could pre-define you vertex "problem group" into a new vertex group (Assign to Group), so that you perhaps have less trial/error or increase your accuracy of selecting the "paste to" group.