Here is the full list of all 49 Star Trek achievements.
After the brawl, escape in less than 6 minutes in "The Trials" chapter
Completed the Gorn Planet & The Trials chapters on any difficulty
Win the brawl against your buddy in "The Trials" chapter
Cleared 30 mini-games including co-operative hacks, bypasses, and power reroutes
Kill 100 enemies with the Vulcan Repeater
Completed the Volatile Cargo & Flashback chapters on any difficulty
Completed the Frontier Starbase chapter on any difficulty
Kill 3 enemies while your teammate (AI or co-op player) is in a downed state
Kill 50 enemies with the Starfleet Phaser Rifle
Completed the Space Battle chapter on any difficulty
Completed the New Vulcan chapter on any difficulty
Kill or take down an enemy that your teammate (AI or co-op) has stunned
Completed the Helios-1 chapter on any difficulty
Completed the Enterprise Capture chapter on any difficulty
Completed the Mothership chapter on any difficulty
Completed the game on any difficulty setting
Completed the game on hard difficulty setting
Kill 100 enemies with the Captain's Phaser
Kill 50 enemies with the Vulcan Pulse Cannon
Kill 20 enemies with the Starfleet Type III Rifle
Kill 50 enemies with the Ravager
Kill 50 enemies with the Marauder
Kill 50 enemies with the Pillager
Kill 50 enemies with the Railer
Kill 20 enemies with the Striker
Kill 50 enemies with the Arc Driver
Kill 20 enemies with Plasma Grenades
Connect every weapon's alternate fire with enemies
Found half of all audio logs
Found all audio logs
Found half of all research items
Found all research items
Complete one chapter in co-operative play
Complete four chapters in co-operative play
Complete all chapters in co-operative play
Bought all upgrades for the Captain's Phaser
Bought all upgrades for the Vulcan Repeater
Bought all upgrades for the Tricorder
Took down 30 enemies in a non-lethal manner
Performed Mind Meld on 10 different enemies
Revived your teammate (AI or co-op player) 20 times
Complete the Space Battle with a hull integrity of 85% or more
Complete all Optional Objectives
Hack an enemy drone to your side and have it kill another enemy
Order your teammate (AI or co-op) and have him complete 5 bypasses in a row
Kill 3 enemies while in a single downed state
As Kirk, get a Gorn enemy to melee you
As Spock, bleed out during a downed state, leaving Kirk alone and failing the mission
Complete half of all Optional Objectives