I'm going to answer similarly to how I answered here.
Let me preface this by saying that I've been heavily influenced by Gothalion. Gothalion is a cross-platform, hardcore Berserker player, known in my book for hosting and winning Underdome tournaments. He has also written a guide to Blast Master...ing.
You should center skill builds around COMs. Which COMs? Offensive COMs, like Bombardier, Blast Master, Skirmisher, Ogre, Bad Ass Torgue, Common Man Tediore, Warmonger, or Berserker. I'm not going to focus on every one of the ones that I mention.
Your build also differs based on the situation. The main situations include PvP, Underdome, PvE (story play-through), boss raiding.
By boss raiding, I pretty much mean Crawmerax the invincible. Crawmerax is only weak to critical hits, and not elements. I practically never raid with Brick, so it's up in the air.
You should use Blast Master for general and most play, Ogre for when the shotgun challenge comes up in the Underdome, Skirmisher for PvP and fun, Bombardier MAYBE for Horde wave in Underdome, and Bad Ass Torgue and Common Man Tediore maybe for other weapon challenges in Underdome, like for use with Torgue and Tediore Anarchies in SMG challenges.
The Brawler tree sucks:
I'll admit, and not reluctantly, that Berserking was awesome. Forget needing to stay on top of the sort of gear that you should have for your level, and just swing those fists, but at the upper levels/end-game...
Mêlée is negligible. This is a case of linear mêlée damage, quadratic bang bang. Although mêlée damage may not necessarily scale linearly, it's still bringing a knife to a gun-fight. At the upper levels, you should only use your mêlée attack offensively if you know the enemy's health is low enough that it will finish them off and you need to reload. Or it's a single enemy, but with a lot of health, and you've got Bash, so you want to Daze them to mitigate their damage output.
When it comes to Berserking, to twist a man's words, "use Berserk never to deal damage, only defensively. Need a heal[?] Pop it." Fact is, even if you're using a Berserker COM, Berserking as a main fighting style is pathetic at the upper levels/end-game. And the upper levels/end-game is pretty much the only time WHEN you'll be able to get the gear and skills to keep up Berserk all the time. Even with maxed out Sting like a Bee, you won't be batting down baddies like this. In contrast, if you're GUNNING, you can. Now, I'll contradict myself again:
In PvP, mêlée with Brick at the end-game is viable and fun.
Duelling is a favourite of Brick, because it's close-quarters, where Berserking shines, and because players do not have as much health as NPCs. Sting like a Bee gets you in there and moving erratically, causing your opponent to panic. The way the screen shakes from your punches is also unnerving to opponents. Skirmisher with +4 Short Fuse means that you'll have a fresh use of your Berserk action skill ready within seconds of your next duel.
Dazing your opponent is one of the most powerful things that you can do in PvP, so Bash is top tier with +3 or +4 from Skirmish pushing the chance of Dazing your victim to 80% or 90%. The only other abilities that match that chance to Daze are one or two of the Siren's close-quarters skills and a skill of Mordecai's Bloodwing. Even in the latter case, you'll be able to get one or two good hits in before Bloodwing touches you.
This is the only form of combat where it's viable to use something other than an explosive artifact.
In longer distance PvP, Brick's got rocket launchers. Shock Redemption or shock Undertaker if you're lucky, are "cheap". One-shot.
Lastly, Underdome and general play:
These two modes are specced for as one. With the ineffectiveness of Berserking as a PvE offensive in the upper levels/end-game, the rest of the skills are close to a no-brainer. When you're not speccing into Brawler, you can nearly fill out the other two trees, save for two skill points. There are two skills in the other trees that are mêlée-related: Bash and Liquidate. The benefits of Bash have already been mentioned.
When it comes to these forms of play (Underdome and general), Brick's specialty is mainly getting the lead out. A legendary Atlas Ogre machine gun and Blast Master COM (and S&S Crux, Spread Destructor, Cobra, and explosive bouncing betties) makes Brick the Blast Master of gibbing.
It is by mostly by Prizefighter, the last Brawler skill, that I still have (seizure warning) fun with Berserking in general play, especially because everyone else gets (notified of) the cash too.
Caveat: You can't necessarily play a style if you don't have the good enough gear.