Here is the full list of all 63 Doctor Who Infinity achievements.
Eliminate more than 3 humans in one turn.
Complete on turn 3 or before
Eliminate more than 3 Roman Daleks in one turn.
Complete on turn 2 or before
Complete in 4 turns or less
Eliminate at least 5 Daleks in one turn.
Complete in 5 turns or less
Complete in 2 turns or less
Eliminate more than 5 Daleks in one turn.
Clear more than 8 question gems in one turn.
Complete without matching any Larvae Gun gems
Complete in 3 turns or less
Eliminate both Zarbis in one turn.
Complete without losing hit points.
Clear more than 9 question gems in one turn.
Complete in 7 turns or less.
Clear more than 12 question gems in one turn.
Clear more than 12 seaweed gems.
Eliminate 2 Ogrons in 1 turn
Complete in 5 turns or less.
Complete 50 levels
Complete 100 levels
Complete 150 levels
Complete the Dalek Invasion of Time
Complete The Orphans of the Polyoptra
Complete The Lady of the Lake
Complete Silent Streets of Barry Island
Complete The Horror of Flat Holm
Eliminate 2 Ogrons in one turn
Clear more than 4 moth gems in one turn
Complete in 1 turns or less
Complete in 2 turns
Clear more than 6 wood pile gems in one turn
Eliminate more than 3 Cormorants in one turn
Complete without matching any Cyberbunny gems
Complete without losing hit points
Complete with 10 or more hit points
Complete without losing any UNIT gems
Complete in 4 turns
Eliminate more than 3 Cyberman in one turn
Complete without losing any tentacle gems