Here is the full list of all 36 MechWarrior Online Solaris 7 achievements.
Kill 10 opponents with Headshots
Get 2500 assists in Light 'Mechs
Get 2500 assists in Assault 'Mechs
Get 2500 assists in Heavy 'Mechs
Get 500 assists in Medium 'Mechs
Get 5 kills in a match
Get 12 assists in a single match
Destroy 1,000 components
Get 500 assists in Light 'Mechs
Get 500 assists in Assault 'Mechs
Kill an opponent with a Headshot
Do top damage in a single match and get zero kills
Complete a match in 1v1 Solaris
Complete a match in 2v2 Solaris
Get 500 assists in Heavy 'Mechs
Get 10 assists in a single match
Equip a bolt-on to any 'Mech
Get 3 kills in a single match where your health never goes below 90%
Destroy 100 components
Win a match with only one person on your team alive
Get 3 kills in a match
Get your first 5 victories in Trial 'Mechs
Destroy 10 components
Get your first victory
Get 5 assists in a single match
Complete the basic tutorial
Get 2500 assists in Medium 'Mechs
Get 8 kills in a match
Earn a sponsor in 1v1 Solaris
Earn all 10 sponsors in a single divison in 1v1 Solaris
Win 5 matches in 1v1 Solaris
Win 15 matches in 1v1 Solaris
Win 50 matches in 1v1 Solaris
Win 100 matches in 1v1 Solaris
Be #1 in any Solaris division after 50 1v1 matches
Equip a bolt-on in every bolt-on slot of a single 'Mech