What are the character specific achievements?

There are 2 achievements per hero, one which unlocks the pixel spray, the other unlocks the cute spray for that hero. You can view them under career profile/achievements. Here is a list of them.

Offensive

Genji

• Their Own Worst Enemy: Kill 2 enemies with a single use of Genji's Deflect in quick or competitive play. Reward: Pixel Spray
• Slice and Dice: Kill 4 enemies with a single use of Genji's Dragonblade in quick or competitive play. Reward: Cute Spray

McCree

• Whoa There!: Interrupt an enemy ultimate ability with McCree's Flashbang in quick or competitive play. Reward: Pixel Spray
• It's High Noon: Get 4 killing blows with a single use of McCree's Deadeye in quick or competitive play. Reward: Cute Spray

Pharah

• Clearing the Area: Knock an enemy to their death using Pharah's Concussive Blast in quick or competitive play. Reward: Pixel Spray
• Death From Above: Kill 4 enemies in a row without touching the ground as Pharah in quick or competitive play. Reward: Cute Spray

Reaper

• Waste Not, Want Not: Get 3 solo kills with a single clip of Reaper's shotguns in quick or competitive play. Reward: Pixel Spray
• Die Die Die... Die: Kill 4 enemies with a single use of Reaper's Death Blossom in quick or competitive play. Reward: Cute Spray

Soldier: 76

• Rocket Man: Get 2 killing blows with a single use of Soldier: 76's Helix Rockets in quick or competitive play. Reward: Pixel Spray
• Target Rich Environment: Kill 4 enemies with a single use of Soldier: 76's Tactical Visor in quick or competitive play. Reward: Cute Spray

Sombra

• Hack The Planet: Hack 15 enemies in a single game of quick or competitive play.
  Reward: Pixel Spray
• Power Outage: Hack six enemies with a single EMP in a game of quick or competitive play. Reward: Cute Spray

Tracer

• Total Recall: Recover 400 health using Tracer's Recall without dying in quick or competitive play. Reward: Pixel Spray
• Special Delivery: Stick 4 of Tracer's Pulse Bombs onto enemies in a single quick or competitive play game. Reward: Cute Spray

Defensive

Bastion

• Triple Threat: Kill 2 enemies in each of Bastion's Configurations without dying in quick or competitive play. Reward: Pixel Spray
• Charge!: Kill 4 enemies with a single use of Bastion's Configuration: Tank in quick or competitive play. Reward: Cute Spray

Hanzo

• Simple Geometry: Get 2 killing blows with a single use of Hanzo's Scatter Arrow in quick or competitive play. Reward: Pixel Spray
• The Dragon is Sated: Kill 4 enemies with one of Hanzo's Dragonstrikes in quick or competitive play. Reward: Cute Spray

Junkrat

• Mine Like a Steel Trap: Knock an enemy into your Steel Trap using Junkrat's Concussion Mine in quick or competitive play. Reward: Pixel Spray
• Roadkill: Kill 4 enemies with a single use of Junkrat's RIP-Tire in quick or competitive play. Reward: Cute Spray

Mei

• Ice Blocked: Block 1200 Damage with a single use of Mei's Ice Wall in quick or competitive play. Reward: Pixel Spray
• Cold Snap: Freeze 4 enemies at once with Mei in quick or competitive play. Reward: Cute Spray

Torbjörn

• Armor Up!: Have one of Torbjorn's Armor Packs on 5 allies at the same time in quick or competitive play. Reward: Pixel Spray
• Raid Wipe: Kill 4 enemies during a single use of Torbjörn's Molten Core in quick or competitive play. Reward: Cute Spray

Widowmaker

• Did That Sting?: Kill 4 enemies using Widowmaker's Venom Mine during a single quick or competitive play game. Reward: Pixel Spray
• Smooth As Silk: Kill an enemy with a Scoped Headshot while airborne as Widowmaker in quick or competitive play. Reward: Cute Spray

Tank

D.Va

• Shot Down: Prevent 1500 damage with a single use of D.Va's Defense Matrix in quick or competitive play. Reward: Pixel Spray
• Game Over: Kill 4 enemies with a single use of D.Va's Self-Destruct in quick or competitive play. Reward: Cute Spray

Reinhardt

• I Am Your Shield: Block 8000 damage with Reinhardt's Barrier Field without dying in quick or competitive play. Reward: Pixel Spray
• Storm, Earth and Fire: Land Reinhardt's Fire Strike and Charge after an Earthshatter Stun in quick or competitive play. Reward: Cute Spray

Roadhog

• Giving You The Hook: Interrupt an enemy Ultimate ability with Roadhog's Chain Hook in quick or competitive play. Reward: Pixel Spray
• Hog Wild: Knock 2 enemies to their deaths with one use of Roadhog's Whole Hog in quick or competitive play. Reward: Cute Spray

Winston

• Mine Sweeper: Destroy 10 Turrets or Traps using Winston's Tesla Cannon without dying in quick or competitive play. Reward: Pixel Spray
• Anger Management: Damage 6 enemies a single use of Winston's Primal Rage in quick or competitive play. Reward: Cute Spray

Zarya

• Power Overwhelming: Keep Zarya's Particle Cannon above 70 Energy for 60 seconds in quick or competitive play. Reward: Pixel Spray
• The Power of Attraction: Capture 5 enemies in a single use of Zarya's Graviton Surge in quick or competitive play. Reward: Cute Spray

Support

Lúcio

• The Floor is Lava: Get 3 killing blows while wall riding as Lúcio without dying in quick or competitive play. Reward: Pixel Spray
• Supersonic: Block 1000 damage with a single use of Lúcio's Sound Barrier in quick or competitive play. Reward: Cute Spray

Mercy

• Group Health Plan: Restore 200 health for 5 players without dying as Mercy in quick or competitive play. Reward: Pixel Spray
• Huge Rez: Resurrect 4 players at once with Mercy in quick or competitive play. Reward: Cute Spray

Symmetra

• The Car Wash: Hit an enemy with 7 beams simultaneously as Symmetra in quick or competitive play. Reward: Pixel Spray
• Huge Success: Teleport 20 players in a single quick or competitive play game as Symmetra. Reward: Cute Spray

Zenyatta

• Rapid Discord: Get 4 kills or assists with Zenyatta's Orb of Discord within 6 seconds in quick or competitive play. Reward: Pixel Spray
• The Iris Embraces You: Restore 1500 health with a single use of Zenyatta's Transcendence in quick or competitive play. Reward: Cute Spray

Source