Family Brawler

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[guide=5828 domain=www]/family-brawler-achievement-guide
If you have any questions please ask on my guide page as I check that more frequently than my comments.
HERE'S A PREVIEW VERSION BELOW(No tips/just pets because word limit):
The guide above will cover building a team/strategies/strengths and weaknesses of each family.)

-Aquatic

Team I completed this with: Frog (S/S Speed 325) / Puddle Terror / Emperor Crab (P/P)

Advised Pets:
Puddle Terror Punch/Nature ward/Dive
Any Frog or Toad(S/S) Tongue Lash/Cleansing Rain/Frog Kiss
Emperor Crab Surge / Healing Wave / Shell Shield

Other mentions:
Snarly
Chuck
Emerald Turtle
Magical Crawdad
Left Shark
Hydraling
Slithershock Elver (S/S)
Spawn of G'nathus
Tottle

-Beast

Team I completed this with: Artic Fox (S/S)/Xu-Fu/Corgilicious

Advised Pets:
Any kind of Fox (S/S)Flurry/Crouch/Dazzling Dance
Xu-Fu Spirit Claws/Feed/Prowl
Corgilicious Bark/Superbark/Puppy Parade
Stunted Direhorn (Trihorn charge/Horn Attack/Trihorn Shield)
Direhorn Runt (Trihorn charge/Horn Attack/Trihorn Shield)
Pygmy Direhorn (Trihorn charge/Horn Attack/Trihorn Shield)

Other Mentions:
Ban-Fu
Feline Familiar
Scalded Basilisk Hatchling (S/S)
Zandalari Kneebiter
Zandalari Anklerender
Zandalari Footslasher
Summit Kid
Alterac Brew-Pup
Son of Skum
Grumpy
Argi

-Critter

Team I completed this with: Rabbit(S/S)/Rabbit(S/S)/Frostfur Rat(P/P)

Advised Pets:
Rabbit/Bunny/Hare(Always S/S breed or 357 speed)
Moveset: Flurry / Dodge / Burrow
Flurry(if faster than 310 speed)Sneak Attack(if slower than 310 speed)/Refuge/Call Darkness
Snail/Conch/Whelk (H/H) Absorb or Ooze Touch/Shell Shield/Dive(This pet offers some ways of dealing with elementals, flying and humanoids)

Other mentions:
Mr. Wiggles and Bush Chicken (for Clone dance - uncanny luck + cyclone)
Lovebird Hatchling
Crispin
Nuts
Armadillo Pup
Black Lamb
Pygamy Cow
Red Cricket

-Dragonkin

Team I completed this with: Spirit of Competition(wouldn't advise -> I just like using this pet)/Lil' Tarecgosa/Chrominius

Advised Pets:
Twilight Clutch Sister Twilight Fire/Phase Shift/Twilight Meteorite
Emerald Proto-Whelp Breath/Emerald Presence/Emerald Dream
Lil' Tarecgosa Breath/Surge of Power/Arcane Storm
Nexus Whelpling (x3) (Tail Sweep/Mana Surge/Arcane Storm (see Tips for basic strategy)
Chrominius Bite/Howl/Surge of Power (Don't be a fool and try Howl->Surge of Power unless you're sure it'll connect, most will switch pets/use a dodge move)

Other Mentions:
Azure Whelp (S/S)
Bronze Whelping
Stormborne Whelpling
Spawn of Onyxia
Onyxian Whelpling
Infinite Hatchling
Dream Whelpling
Yu'la, Broodling of Yu'lon
Sprite Darter Hatchling
Nether Faerie Dragon

-Elemental

Team I completed this with: Nightshade Sproutling (S/S)/Molten Corgi/Stormstruck Beaver

Advised Pets:
Nightshade Sproutling (S/S or 325 speed) Lash/Call Darkness/Blinding Poison(Speed is key with this pet as all of these moves do better if they go first.)
Stormstruck Beaver Tail Slap/Lightning Shield/Discharge
Molten Corgi Bark/Superbark/Puppies of the Flame
Lil' Ragnaros Sulfuras Smash/Magma Trap/Sons of the Flame
Terrible Turnip Weakening Blow/Leech Seed/Sons of the Root
Broot/Blossoming Ancient Ironbark/Photosynthesis/Sunlight

Other mentions:
Pandaran Water/Earth/Air/Fire Spirit
Jademist Dancer
Droplet of Y'shaarj
Singing Sunflower Solarbeam//Photosynthesis/Sunlight
Stout Alemental
Skunky Alemental
Crazy Carrot
Ruby Droplet

-Flying

Team I completed this with: Junglebeak (P/P)/Dread Hatchling/Stormwing(just because I like using her :P)

Advised Pets:
Northern Hawk Owl/Long-Eared Owl/Fledgeling Warden Owl Quills/Darkness/Nocturnal Strike
Dread Hatchling Shadow Talon/Darkness/NocturnalStrike
Crow/Raven Alpha Strike/Darkness/NocturnalStrike
Teroclaw Hatchling (A great pet if you don't want to go the Glass Cannon route -> Always open with Nature ward then dodge -> retreat when low health in Nature's ward is up) Alpha Strike/Dodge/Nature's Ward)
Moths Cocoon strike/Moth dust/Alpha strike
Stormwing Quills/Lightning Storm/Thunderbolt
Jungle/Blood/Axebeak (P/P) Peck/Rain Dance/Nocturnal Strike

Other Mentions:
Chi-Chi
Direbeak Hatchling
Bloodgazer Hatchling
Snowfeather Hatchling
Sharptalon Hatchling
Royal Peachick
Ji-kun Hatchling
Any Gryphon

-Humanoid

Team I completed this with: Anubisath Idol/Bonkers(S/S)/Horde Fanatic(or Alliance Enthusiast if you're Alliance)

Advised Pets:
Anubisath Crush/Sandstorm/Deflect
Qiraj Guardling Crush/Sandstorm/Blackout Kick (same strategy as above)
Bonkers (S/S or 325 speed) Bite/Dodge/Tornado Punch (speed is important as Dodge going first ensures an extra invulnerable turn and if Tornado Punch procs it's stun the enemy doesn't get it's move off)
Horde Fanatic/Alliance Enthusiast Ban Hammer/Jar of Smelly Liquid/BURRRP!
Kunlai Runt Takedown/Frost Shock/Deep Freeze (open with Frost Shock into Deep Freeze then takedown)
Grumpling Punish/Frost Shock/Deep Freeze (same strategy as above)
Fiendish Imp (S/S or 333 speed) Burn/Immolation/Nether Gate
Murkalot (if you have this USE IT, ALWAYS LEAD WITH IT AND USE RIGHTEOUS INSPIRATION INTO A PET WITH A HEAVY HITTING MOVE)) Punish/Falling Murloc/Righteous Inspiration
Ore eater Acid Touch/Shell Armor/Bodyslam (always have shell armor up before you Bodyslam)

Other Mentions:
Wyrmy Tunkins
Harbinger of Flame
Flayer Youngling (S/S)
Father Winter's Helper
Curios Wolvar Pup
Tylarr Gronnden

-Magic

Team I completed this with: Eldritch Manafiend/Arcane Eye/Arcane Gorger

Advised Pets:
Arcane Eye: Psychic Blast/Drain Power/Mana Surge
Arcane Gorger/Mana Wyrmling Flurry/Drain Power/Mana Surge
Eldritch/Erudite Manafiend Arcane Explosion/Surge of PowerArcane Storm
Servant of Demidos Magic Sword/Siphon Anima/Soul Rush
Nordrassil Wisp (S/S) Light/Flash/Soul Ward

Other Mentions:
Void Shardling
Jade Oozeling
Ethereal Soul-Trader
Minfernal
Brightpaw
Stardust
Twilightlight Fiendling
Trunks
Hyjal Wisp
Hungering Claw
Darkmoon eye

-Mechanical

Team I completed this with: Clockwork Gnome/Mechanical Pandaren Dragonling/Tranquil Mechanical Yeti

Advised Pets:
Clockwork Gnome Railgun/Blitz/Build Turret
Tranquil Mechanical Yeti Thrash/Call Lightning/Ion Cannon
Mechanical Pandaren Dragonling: Breath/Thunderbolt/Decoy (Literally three of these is an easy way to cheese this)
Iron Starlette Wind-up/Powerball/Supercharge (Wind-up+Supercharge+Wind-up= 1-shot KO BUT if an opponent is smart they will try to stun/block this)
Lil' Bling Inflation/Extra Plating/Make it Rain

Other mentions:
Cogblade Raptor
Darkmoon Tonk
Darkmoon Zepplin
Ancient Nest Guardian
Fluxfire Feline
Personal World Destroyer
Lifelike Mechanical Frostboar
Clockwork Rocket Bot
Son of Animus
Sky-bo

-Undead

Team I completed this with: Unborn Val'kyr(H/H)/Bone Serpent(H/H)/Fossilized Hatchling

Advised Pets:
Unborn Val'kyr (H/H) Shadow Shock/Curse of Doom/Haunt
Fossilized Hatchling Claw/Death and Decay/Bonestorm
Wicked Soul (S/S) Shadow Shock/Haunting Song/Haunt
Ghastly Kid Hoof/Ethereal/Haunt
Weebonimination Cleave/Consume Corpse/Haymaker (or corpse explosion)
Graves Skull Toss/Grave Destruction/Consume Corpse
Bone Serpent (H/H) (x3) Bone Barrage/Call Darkness/Nocturnal Strike

Other mentions:
Blight Hawk
Blightbreath
Scourged Whelpling
Ghastly Rat
Ghostly Skull
Frostwolf Ghostpup
Creepy Crate
Mr. Biggleworth
Son of Seethe
Fragment of Anger
Fragment of Desire
Fragment of Anger
Fragment of Desire
Fragment of Agony

(Seriously, any undead pet is viable)

Just got this achievement done over the weekend, I'll post my teams and will keep it simple... you guys are studs and can see what I was going for.

Aquatic (lost a lot)
Left Shark
Snarly
Crab p/p with blood in the water
(Goal is to duel with left shark and speed up team before he dies...)

Undead
Unborn valk
Fragment of anger
Graves
Easiest one by far. Just dot em up and win. Consume corpse with graves x2 is op

Critter
2 rabbits s/s breed
Zoom
Basically just duel with rabbits finish off with zoom if needed

Beast
Fox w/ S/s
Rose taipan
Xufu (ideal to use after you howl with fox for bit bites)

Mechanical (10-0 with this team)
Blingtron (Use the100% dmg debuff ability)
Mechanical Pandaren Dragonling
Iron Starlette (bait dodges etc with Powerball)

Magic (most fun team imo)
Son of Demidos
Hyjal Wisp
Blood Boil

Elemental
Ragnaros
Broot
Pandaren Water Elemental
(Just kinda relied on rag to wear em down with stuns and wave etc... but if he got toasted early I either lost or had a long duel with broot)

Flying (10-1 here, strong team)
Ikky (for debut)
Gilnean Raven
Warden owl (basically 2 birds with darkness)

Humanoid (admittedly harder for me than most of the others)
Squirky (for debuff)
Kun-Lai Runt
Flayer Youngling (I may have used another, don't quite remember this was first one I went for and swapped a lot)

Dragon (about 55% win rate here, once I had a better strategy it helped... end games with chrom don't start lol)
Bronze whelping
Emerald Proto whelp or nexus whelping
Chrominius

This is a good achievement to work on when it is a Pet Battle Bonus Event week. This is why:

• 
  
• The achievement is account wide, so you can complete the quest The Very Best with any of your alts who are high enough to get that quest, rewarding you with Ultimate Battle-Training Stones on each toon you complete the quest on.
• 
  
  
• The ques / waiting times for battles tend to be shorter during the Pet Battle Bonus event, since many people are trying to complete that quest.
• 
  
  
• The people who are queing, in general, are not the same "hard-core pet battlers" (oxymoron?) guys who are queing when its not a pet battle event, so you are more likely to find people with teams you can actually beat.
• 
  

I categorize this achievement as a "building a mountain a spoon-full of dirt at a time" kind of achievement, I completed it over a long period (several pet battle bonus event weeks), I found I got too frustrated with it if I did it for too long. Good luck!

-- Edit --

Sometime after making this post Blizzard nerfed the weekly quest so you can now only do it once per account (very frustrating because all other weekly quests are alt friendly, why not this one?, and because it has dramatically reduced the number of teams queuing. Please revert this change Blizzard!)

Also... once you have completed this achievement, have some sympathy for those still doing it. When I am queuing for pet battles and I keep getting matched with some guy trying to get his all aquatic team the wins he needs, I will often forfeit some of the games to him. I remember how painful this grind was.