To get this achievement, you need to win the Constructors' Championship in career mode on any difficulty.
The Constructors' championship is basically like the drivers championship, but its your racing team's points (your's and your teammates) that add up to the overall point total of the team.
It doesn't matter who you start with, since you have the option to switch to one of the top 3 teams after race 9 or 10 if you perform well enough (I was offered a contract by Mercedes AMG Petronas). Just make sure you start with a decent team. I suggest unlocking either Sauber or Lotus via Young Drivers Test. I started with Lotus and won 8/10 first races on easy, so it is not difficult to rack up points, but you need to do well for one of those three teams to offer you a contract.
Also, don't worry about how many points you obtain for your starting team. With Lotus, I had them around 270 points by race 10 before I switched to Mercedes AMG. But since Lotus and the other teams rarely finish in the top 10, and the other three teams always finish top 5, you surpass your old team very quickly. I won every race with Mercedes and since my teammate did pretty well as well, I was able to unlock the achievement right after the USA grand prix, which is the second last one. I finished about 100 points clear of the second place team and I didn't even bother racing the last grand prix.
If there's anything else I should add, let me know.