For this you must defeat all 24 characters from the game in succession, with your health bar and Super meter carrying over. If you lose you will be taken back to the main menu, no retries allowed. The difficulty setting for the opponents vary from Very Easy to Hard in this mode. You do in fact get 25% or more of your health replenished after each fight depending on how well you fought. An extra 10% of health will be awarded to you for each bonus objective you complete. After each match you win, the game will tell you your bonuses during your win pose. Try to complete some of the following objectives to get as much Health bonuses as you can:
Win the match: 25%
Perform a 30% Combo: 10%
Use an Interactive object: 10%
Win a Clash: 10%
Complete a Transition: 10%
This Battle challenge isn't too difficult actually, but you'll have to play smart and careful with your health meter. Use the above mentioned Doomsday method
for this, since Hawk Girl has too many bad match ups and will take too long to win fights spamming Mace Charge. The first 10 opponents or so are usually very easy to defeat, but somewhere toward the middle you'll face an opponent who won't fall for your combo so easily. Try crouch blocking against them if they are aggressive and use the pause button until they open them self up then punish. Personally, I found the crouching heavy attack(Down +A) to work well up close.
You want to be smart with your Health and Super Meter because both of these carry over to the next match. Never blow your Super meter on a Super Move, your combo is good enough. Only use meter for Meter Burn with your Venom Tackle or to win clashes, that's it. More than likely while on their 2nd Health Bar(red health), the CPU will try to Clash you when you land a combo on them, so more meter increases your chances of winning or tying those clashes to prevent health loss. Remember, that using single hit heavy attacks(A-button) help prevent the CPU from clashing against you, but then you'll have to worry about them trying to hit you with their super move. Also use interact-able objects whenever you can, and keep sticking to the combo. Only start pausing against the more resilient CPU and scout for openings. Focus on keeping your health as high as you can and you'll get through all 24 foes. Just make sure you have at least a little over a red health bar going into fights. For me, I had just a sliver of white health going into most fights and I was still able to finish strong. Buckles Supreme has a video of this being done below.
Hopefully these methods prove effective for you. Please comment if anything needs to be added or changed.