Here is the full list of all 14 SPECTRE achievements.
You started playing Spectre!
You hacked an objective as a Spectre
Knock out a Reaper with a RIGSY
Snap a Reaper's neck
Perform a co-operative move
Kill two Reapers in the same round
Kill two different Spectres with headshots in the same round
Kill two Spectres with one explosive
Headshot a Spectre while they are actively hacking an objective
Accumulate a total of 500 kills as a Reaper
Win a round with 10% or less of your health remaining
Win a game (all rounds) without sustaining any damage
Win both rounds of the match
Win a third round