Cube character rigging - the next steps...?

Hi,

I'm looking to rig my little characters so that they deform and move similar to the behaviours in this video. (this question currently only refers to information regarding the cube character)

https://www.youtube.com/watch?v=i90ZMD64iVg

These little guys have been my initial character creations since I began my journey into 3D animation. They were originally modelled and animated in C4D using simple deformers over a year ago before I had to pause my 3D journey.

I have since switched over to Blender the past few months, and am looking to create a rig that will allow me to control these characters beyond the simple deforms. I have followed the "fundamentals of rigging course" and have a basic understanding of the tools involved, and have successfully managed to apply a "bouncing ball" rig to the cube body (minus limbs). However there is a distinct lack of stretching and bulging in the mesh.

Could anybody please point me in the right direction of how to recreate a similar effect to the cube deforming? Should I be using shape keys? (this is high on my list of tutorials as I've never done it before) does it depend on weight painting? (still awaiting the new lesson as this question is posed). Am I wrong to assume using an armature and constraints alone is enough to create the effect I'm looking for?

I'm a big fan of problem solving, and not afraid of a challenge, but being new-ish to Blender (<1yr) and brand new to rigging (<1week) has left me wondering where to go now... I would appreciate any direction pointing in which I should be expanding my knowledge to give these little guys their groove back! Thanks so much!