Fight challenges are tracked in the fight challenges menu. I do not recommend grinding for fight challenges until you have completed all the other achievements. You will most likely earn this achievement on the way.
There are 22 different things that are tracked for Fight Challenges. They are leveled 0-10. They are as follows:
Special Move 1
Special Move 2
Special Move 3
Auto Double Hits
When grinding for fight challenges I recommend doing classic survival mode. You can do vs matches but it requires more inputs to retry each match, survival automatically moves on to the next match.
When in a match the best combo I found was starting with a jump in attack. I would then push a button and immedietly counter break. I would then mash out an auto combo. If I had shadow meter I would use it in the combo, and the I would do an ender to end the combo.
What this would do for the fight challenges is: the jumping attack would count as an Aerial attack and an Overhead. The next hit would count as a Manual and if the Counter Break was successful it would count as one of those. Each hit after that would count as an Auto Double, if using a Shadow move it would count, and then the Ender would count as well. The entire combo would count for Combo Hits.
Once the opponent was low enough I would end in an Ultra, and make sure you let the entire Ultra Combo go because each hit of the ultra counts.
This would then add one to Supreme Victory, Survival match, and Ultra use.
For Spinal specifically I would do his jumping Heavy Kick and follow up with a standing Heavy Kick to start the combo. For an Ender I would do the Bone Sword to get counts for Juggles, or I would use the Boneshaker to get a wall splat to follow up with a Throw. If I had meter I would do the Shadow Skeleport because it counts as both Aerial and Juggles.