Here is the full list of all 19 Invasion achievements.
10 deaths in row
Kill 500 ufos
Kill self by nuke
Kill 100 ufos
4-9 kills per time
Kill self
Firing accuracy <= 10%
Kill 2k ufos
Kill 10k ufos
Kill 50k ufos
10-20 kills per time
21+ kills per time
Firing accuracy >= 95%
Firing accuracy == 100%
Finish level with 1hp
Finish level without damage
Use 9+ nukes per level
Use 21+ nukes per level
Use 666+ nukes