Here is the full list of all 128 BATTLETECH achievements.
Destroyed 5 enemy vehicles with melee
Destroyed 5 enemy 'Mechs in melee
Completed your first mercenary contract
Refitted a 'Mech with only Energy weapons
Suffered a terrible betrayal
Removed both arms from an enemy 'Mech
Removed one arm from an enemy 'Mech
Destroyed an enemy 'Mech with an Alpha Strike
Made a melee attack on an enemy 'Mech that was knocked prone
Removed both arms and one leg from an enemy 'Mech
Established a history
Recovered an ancient ship
Started a war
Liberated the world of Panzyr
Liberated the world of Smithon
Captured a villain
Discovered ancient technology
Liberated the world of Itrom
Learned an important secret
Liberated the world of Tyrlon
Invaded a capital
Crowned a rightful ruler
Battled someone who has this achievement (and therefore has either battled HBS developer/BT tourney champ Chris Eck or battled someone who battled someone...who has)
Played 5 online multiplayer matches
Played 50 online multiplayer matches
Played 100 online multiplayer matches
Won an online multiplayer match
Won 25 online matches
Won 50 online matches
Won 100 online matches
Lost 10 online matches
Won a match with three 'Mechs vs. an OpFor with four 'Mechs
Won a skirmish match at the Clash (15m) Lance Value Budget
Won a skirmish match at the Battle (20m) Lance Value Budget
Won a skirmish match at the War (25m) Lance Value Budget
Won a skirmish match with more than 25m Lance Value on both sides
Won a skirmish match using fewer 'Mechs than the AI opponent
Won a skirmish match with a Lance Value 5m less than the OpFor's
Won a skirmish match with a Lance Value 10m less than the OpFor's
Won a skirmish match with a Lance Value 15m less than the OpFor's
Won a 4v4 match with all of your 'Mechs being Locusts
Lost to the AI one time
Won 10 skirmish matches
Won 50 skirmish matches
Won 100 skirmish matches
Destroyed 50 'Mechs
Destroyed 250 'Mechs
Destroyed 500 'Mechs
Destroyed 50 assault 'Mechs
Destroyed 50 heavy 'Mechs
Destroyed 50 medium 'Mechs
Destroyed 50 light 'Mechs
Killed a Pilot without destroying their 'Mech
Ejected 5 times
Ejected 20 times
Destroyed an enemy 'Mech's head with a Called Shot
Destroyed an enemy 'Mech's leg with a Called Shot
Missed a Called Shot
Destroyed 5 enemy 'Mechs with Death from Above
Destroyed 20 enemy 'Mechs with Death from Above
Destroyed 50 enemy 'Mechs in melee
Destroyed 50 enemy vehicles with melee
Destroyed an enemy 'Mech and your own 'Mech while using Death from Above
Destroyed two 'Mechs in the same turn using Multi-Target
Destroyed an enemy light 'Mech with a Highlander Death from Above
Destroyed an enemy 'Mech with Death from Above using an UrbanMech
Destroyed an UrbanMech with Flamers
Destroyed an enemy Orion with an AC/20 headshot
Removed all the limbs of an enemy 'Mech
Your light 'Mech killed an assault 'Mech with a melee attack
Fielded a lance of four UrbanMechs, won, and lost no 'Mechs
Fielded a lance of four King Crabs, won, and lost no 'Mechs
Maxed out a MechWarrior's Guts stat
Maxed out a MechWarrior's Tactics stat
Maxed out a MechWarrior's Piloting stat
Maxed out a MechWarrior's Gunnery stat
Hired five MechWarriors
Hired 25 MechWarriors
Hired 50 MechWarriors
Hired 100 MechWarriors
Saved up over 5 million C-bills
Saved up over 10 million C-bills
Saved up over 20 million C-bills
Purchased a new 'Mech chassis
Negotiated for nothing but salvage in a completed contract
Negotiated for nothing but reputation in a completed contract
Completed 10 mercenary contracts
Completed 25 mercenary contracts
Completed 50 mercenary contracts
Completed 100 mercenary contracts
Retreated once from a contract before any of your 'Mechs got destroyed
Retreated 10 times from a contract before any of your 'Mechs got destroyed
Won a skirmish match with three Commandos vs an AI Atlas
Won a skirmish match with an Atlas vs four AI Commandos
Successfully completed a contract using only one 'Mech
Destroyed an enemy 'Mech with Flamers
Have a 'Mech and its Pilot survive being knocked down, shutdown, and shot in the head
Your pilot ignored the safety instructions and destroyed their 'Mech by overheating while firing weapons
Refitted 25 'Mechs
Refitted 50 'Mechs
Refitted a 'Mech with only Ballistic weapons
Refitted a 'Mech with only Missile weapons
Refitted a 'Mech with only Support weapons
Refitted a 'Mech other than an Awesome equipped with three or more PPCs
Completed 25 'Mech repair work orders
Refitted a 'Mech with the maximum amount of armor
Watched all of the credits
Acquired all 36 launch date 'Mechs
Built 10 'Mechs by collecting/salvaging their parts
Completed a Campaign with Iron Man enabled
Start a campaign with 'Mechs requiring 8 parts, then complete 10 'Mechs
Become Allied with House Steiner
Become Allied with House Davion
Become Allied with House Liao
Become Allied with House Kurita
Become Allied with House Marik
Become Allied with the Magistracy of Canopus
Become Allied with the Taurian Concordat
Completed a contract with one mech remaining and no pilots killed
Completed your first Career Mode campaign with any rank
Accomplished a Career Mode score at Veteran rank (>320,000 points)
Accomplished a Career Mode score at Elite rank (>480,000 points)
Accomplished a Career Mode score at Legendary rank (>720,000 points)
Completed Career Mode with Iron Man enabled