The Berserker is typically a more powerful version of a Muton, deprived of a firearm and grenades, focusing solely on melee combat. It has a mind-boggling amount of health, but lacks any armor, which, paradoxically, makes him extremely easy to put down, mainly because of the fact that when he appears on the battlefield your soldiers should be equipped with better (if not the best) gear possible.
The best method to deal with the enemy is to keep your soldiers at a safe distance - which is, more or less, about the range of sprint of a typical soldier. This way the enemy won't be able to run to your soldier and attack him / her. However, dealing damage to a Berserker has a chance to cause him to enter an enraged mode, increasing his mobility. A successful Berserker attack, especially if he manages to land a critical hit, can kill your soldier in one strike.
The enraged mode of the Berserker can be used against him - whenever he enters it, he will attack the closest target, often pounding his own allies, allowing you to just stand and watch as aliens kill each other.
The enemy is extremely vulnerable to the Sharpshooter class, using the Deadeye ability, as well as the Ranger with the Rapid Fire skill - both of those abilities give a penalty to AIM, but given the fact that the Berserker has 0 defense that shouldn't hinder them at all. Both of those attacks should be able to either kill the Berserker in one shot, or severely wound him at least.