A quick and easy method to defeat Top Dogs.
Upgrade Max's Shotgun (Optional)
Upgrade the Shotgun in the "Max" menu. The more you upgrade it, the better. More shots means more chances to shoot. The Top Dog will be down in around 4 shots. However, he can block shots entirely if his arm and weapon are positioned *just so*.
Unlock Shiv Punishers (Optional)
Shiv punishers can be used to give a little extra damage, but you can also get through a fight without them. Unlock the punishers in the "Max" menu under "Skills".
Find a Top Dog
Top Dogs are located at Top Dog camps as well as in a few scavenging locations (surprise!). Locate a Top Dog. It's probably good to start with one that has a lower difficulty rating (fewer skulls in the description).
Clear the Area
The Top Dog will likely release a few War Boys to get in your way. Kill them first using unarmed attacks. Try not to use any shivs and definitely do not use the shotgun. Avoid the Top Dog while you are doing this. Run across the entire fighting floor if you have to. You can also repeatedly "dodge" (RB on the controller if you are using it) to quickly roll away from everyone.
- Run up fairly close to him (default RT on controller).
- Once he's using one of those "enraged" attacks (where a red skull appears above his head), quickly dodge the attack (RB).
- Spin around to face him. He'll likely have stumbled and be fumbling with his melee weapon.
- While he is distracted, quickly aim your shotgun (LB) and shoot him (B).
- While he is stumbling from the shot, run up near him and use a Shiv Punisher (A).
- Rinse and repeat.
- Once his health is very low, run up and finish him off (A).
This is a lot simpler than trying to punch him to death, use the environment, or to keep using melee weapons you've found or shivs that will run out before he dies.