To complete a contract, first you must sign a contract.
This can be done by logging into the game normally, than using the LB/RB buttons to go to "Tanks" tab, selecting the "Mercenaries" nation. After this, find a tank that is avaible (grey = unavailable and white = available). After selecting a tank select "Sign Contract" in the bottom right of the screen.
You can read what is required for all stages of a contract at the screen where it shows the tank. You can look at stages ahead with the left/right D-pad.
When you complete the first stage of any contract, the achievement should unlock.
NOTE - the contract for the tier V may be glitched. (Credit to AVV Enrico)
NOTE 2 - I don't know if the achievement will unlock if the stage is skipped by using gold. If anyone knows, let me know and I'll update the guide, with credit to you of course.