Here is the full list of all 23 Astro Chef achievements.
Complete the tutorial level
Complete Chapter 1 in Single Player
Complete Chapter 2 in Single Player
Complete Chapter 3 in Single Player
Complete Chapter 4 in Single Player
Complete Chapter 5 in Single Player
Complete the Easiest Chapter in Co-op
Complete the Easy Chapter in Co-op
Complete the Medium Chapter in Co-op
Complete the Hard Chapter in Co-op
Complete the Hardest Chapter in Co-op
Fail one Co-op level
Unlock all rewards
Serve a total of 10 dishes to customers
Serve a total of 25 dishes to customers
Serve a total of 50 dishes to customers
Serve a total of 100 dishes to customers
Get a 1 star rating from a customer
Get a 2 star rating from a customer
Get a 3 star rating from a customer
Get a 4 star rating from a customer
Get a 5 star rating from a customer
Complete a shift with a minimum of 3 5-star dishes