Rig the letters with an armature. Make a bone chain for each letter you would like to animate. The complexity of each chain is up to you. I noticed a few letters with “legs” in the video you posted. I’d use a simple IK chain for those letters to accomplish that. You will also need to figure out some squash and stretch controls for each letter. That can be accomplished multiple ways.
Here is a simple demo. (Complete with some very dodgy animation.)
I used Rigify to create this quickly. (You’ll need Auto Run Python Scripts checked in Preferences in order to use the custom Rig buttons.) It’s not intended to be a complete rig, just give you some ideas.
I used the Pitchypoy tentacle rig to auto create a simple squashy bone chain for each letter. Each letter is being deformed by a deform cage (deform modifier, deform meshes are located on layer two.) The deform cages are parented and auto weighted to each bone chain.
I converted the letters from a text object, so the topology is awful.
I hope this gives you some good ideas.
LetterRig.blend (1.6 MB)