To separate your clips by scene version, you must proceed like this:
First, you create a scene and, in this scene, create the full rig for your character, without animating it.
Then, you save a new version of your scene for the first animation clip you want to create. To do this, select File > Save As New Version from the top menu. Give this scene version the name you want your clip to have. For example, if you want to create your character's idle animation, you could name this scene version Idle.
In this scene version, create your character's animation. Make sure the animation starts as soon as the scene begins and that the scene ends as soon as the animation is over.
To create another clip, repeat steps 2 and 3 until you have all the clips you need for your character.
In the end, your scene and its versions may have something like this: