To win the tournament you have to do the following:
Round 1: Win 6 legs (2 sets of 3 legs)
Round 2: Win 9 legs (3 sets of 3 legs)
Quarter Finals: Win 12 legs (4 sets of 3 legs)
Semi Finals: Win 15 legs (5 sets of 3 legs)
Final: Win 18 legs (6 sets of 3 legs)
This tournament is a pain because you have to 'double in' to be able to score.
The way I found the best is:
Bull, Bull, T20
T19, T19, T19
T20, T20, Bull
Bull counts as a double in. I found that out on the second season.
You have to do the whole tournament without leaving the career mode or turning your xbox off as this disables the achievement.