Here is the full list of all 15 DC League of Super-Pets: The Adventures of Krypto and Ace achievements worth 1,000 gamerscore.
Finish a level without destroying a LexBot
Fill PBs Safety Meter
Defeat Lex Luthor
Fully Upgrade Amazon Resistance
Fully Upgrade Dodge Force
Fill Mertons Safety Meter
Defeat the OctoBot
Fully Upgrade your Stamina Meter
Fully Upgrade your Health Meter
Fill Chips Safety Meter
Defeat the SnakeBot
Fully Upgrade Energy Twin
Complete your First Adoption
Finish a level with at least 85 percent health
Achieve a Doggo Rating