You'll need to complete 30 races in the UFO to complete the career. You don't necessarily need to win all the races or championships, but you can try if you want. You need to complete all 3 championships:
S1 - 5 Races
S2 - 10 Races
S3 - 15 Races
I would recommend just trying to get first in each championship. First in all the races will likely be very difficult. The problem is the UFO controls are very fidgety. It's easy to get caught on walls and other racers. You shouldn't be surprised if all the racers all bunch up and can't escape a corner in some races. This means that passing racers can be very difficult.
If you can get 1st a couple times each season, you will easily take first in the championship. Unlike other racer games with championship modes, racers do not trend towards a single position. In other words, the racer who finishes second to you in one race likely won't be there the next one. It is more likely that they finish towards the back of the pack. This is the result of the following:
Start order is determined by rankings rather than the previous race's finishing order. Last place starts first in the race. This means that the racers closest to you in the standings (if you're in first), will be starting at the back of the pack like you. In general, the computer controller racers starting in last do not pass the people in front them.
Achievement unlocks after finishing the 3rd UFO season.