Easiest way for a Blender beginner to do what you want is to import the animations fbx and the character you want in the same blend file. Then on the character you want to use, to each part of the character you add a new 'Object Constraint' and set it to 'Child Of'. Then you select the Armature of the fbx as your 'target' (that would be the KayKit Animated Character .fbx) and under 'Bone' you select the corresponding piece of the character (head to head, armLeft to armLeft etc.)
It's best to have both the characters located at 0,0,0 postion, so that the pieces line up by default. If they don't you need to press 'Set Inverse' in the newly created object constraint on the pieces that are not aligned.
After this you have the 4 pieces of the new character (head,body,armLeft,armRight) connected to the skeleton with all the animations, you can now just delete the Prototype Pete mesh and you have your new rig set up!
Let me know if this worked for you or if you have any questions.