Elemental Brawler

That was a horrible achievement. In the end, 3 pets with conflagrate and burns was my most successful team.

Fel Flame (2,1,1)
Dark Phoenix Hatchling (2,2,1)
Lil' Ragnaros (1,2,1)

But any 3 pets with conflagrate and some kind of burn, preferably Scorched Earth, might be a good choice.

I found one team and made another, both of which were fun and won a lot more than they lost.
Credit for Team 1 should go to Rosqo Pet Battler, on YouTube, who has made a series of videos that cover team compositions for the various component achievements that make up the Family Brawler achievement.

---

Team 1 - Clone Dance:
Droplet of Y'Shaarj (P/P, (2,1,1: Swallow You Whole, Acid Rain, Dreadful Breath)) - Source: Sha of Pride (Drop, Siege of Orgrimmar (Flexible, Normal or Heroic)), Auction House
Drafty (S/S, (1,2,1: Wild Winds, Cyclone, Autumn Breeze)) - Source: Nezir (Drop, Throne of the Four Winds), Auction House
Jademist Dancer (P/S, (1,2,1: Steam Vent, Rain Dance, Acid Rain)) - Source: Jademist Dancer (Drop, Timeless Isle), Auction House

Strategy:
Drafty: Cyclone, then switch pets if possible. If you cannot switch (I was frozen for two turns, once, by another elemental team), then Autumn Breeze and Wild Winds until you can.
Jademist Dancer: Rain Dance, Acid Rain, Steam Vent, Steam Vent until death. With Acid Rain up and Cyclone bouncing around, I finished many teams with Steam Vent alone. It hits hard, even without boosts, and the P/S Jademist Dancer is a touch faster than several pets.
Droplet of Y'Shaarj: Re-apply Acid Rain if needed. Here you have the luxury of options. I often found that the opponent had no more back-line pets, or that those pets had been reduced to extremely low health. Rather than go with Dreadful Breath - which I would if the opponent's back-line pets had not been weakened enough, I found a lot of satisfaction in the savagery of Swallow You Whole. It's often able to finish teams.

Comment(s):
This team is reliable and can polish off many teams with ease but, as is the case with most single-family teams, can't do everything all the time. It requires a bit more attention than Team 2. (Clone Dance teams tend to have that name because of Cyclone and because the original versions of the team have been so often successful and copied.)

---

Team 2 - Traps:
Blazehound (P/P, (1,2,2: Bark, Frenzy, Magma Trap)) - Source: Shannox (Drop, Firelands), Auction House
Magma Rageling (P/P, (1,1,2: Flamethrower, Flame Jet, Magma Trap)) - Source: Ignis the Furnace Master (Drop, Ulduar), Auction House
Lil' Ragnaros (P/P, (1,1,2: Sulfuras Smash, Magma Trap, Sons of the Flame) - Source: Blizzard Store (in-game / in-browser)

Strategy:
Blazehound: Magma Trap, Frenzy, Bark. Alternate between Frenzy and Bark until death, adding Magma Trap if / when possible.
Magma Rageling: Magma Trap, Flame Jet, Flamethrower. Keep up Flamethrower until death, adding Flame Jet and Magma Trap if / when possible.
Lil' Ragnaros: Magma Trap, Sulfuras Smash. Keep Sons of the Flame for when something needs to be avoided. Otherwise, keep up Sulfuras Smash and add Magma Trap if / when possible.

Comment(s):
The idea for this one came from an existing mixed-trap team (Curious Wolvar Pup (2/1,1,1) and its trap is in there, instead of Blazehound, and the pet order is reversed).
This is not a complicated team, which makes it a great alternative / cleanser for the other one. It can also be more easily played while your attention is divided.
When this doesn't win, because it has several attacks that you don't see every day, it's still fun. When it does win, it's fantastic. To watch those traps explode, stun and (often) take out your enemies is a thing of beauty. Traps are so good that it's possible to beat teams that are strong against your elementals.

I had relatively good success with:

Fel Flame 1/2/1
Ominous Flame 1/2/1
Ruby Droplet 2/1/2

The general strategy was to cast Scorched Earth followed by Conflagrate.

Then - depending on the enemy response - either finish their first pet with Burn or two or switch to the other Flame and cast Conflagrate again.

The Droplet is fine for mopping up - with healing either with Absorb or Drain Blood.