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