Win the LMP1 World Championship withtin ten seasons.
1) Start campaign.
2) Select tier 8 on the far left of the tier selection screen. This is a requirement.
3) Play through your first season.
4) At the end of your first season you should have some contract offers open up. When accepting a contract offer, try to accept the one that sounds like the biggest step up. Just note that sometimes all of your offers will be from the same series, so in that case it doesn't matter which one you accept.
5) After a few seasons, you should start to get contract offers from teams in the LMP series. Keep an eye out for the keyword LMP3 in your mail. After winning 1-2 LMP3 championships, you should get an LMP2 offer. And finally after winning 1-2 LMP2 championships you should get the LMP1 offer that you've been waiting for. Just win this championship and the achievement should be yours.