This is quite easy to do and you don't have to rely on other people for it.
First you have to create a second PSN account or load it up if you already have one. Start the game and play the introduction race. Now quit the game and log out with your second to start up your main. Play a Single Event with Drift being the shortest. Nilgiri Hills (India) and Holmastad (Norway) are nice and short. Get a good amount of points because you can't avoid getting some speed points with your second in a few minutes. Use a level 1 car (Mini is probably the best choice) since you can't play the challenges with your secondary account if you use a high level ride.
After you finished the race go back to the main menu. Select
Challenges -->
Solo -->
Create. The first entry in the list is your last race. Set the timer to 1 hour (has been changed from 30 minutes recently) and make it Invite Only. Go to
Add Recipients and the friends folder. Pick your second account and confirm. Send it. Do this 4 more times with the same race (they don't have to be different events).
Next you'll switch back to your secondary. Go to
Challenges -->
Solo -->
Received. Here you'll find your main's challenges. Drive all 5 of them and make sure
NOT to beat the score.
After you're done just go back to your main and wait for the time to run out. You can see how much time is left in the Ongoing section of the Challenges. You might also wanna wait in said Ongoing section for the time to run down because otherwise the Trophy might not unlock right away.
Should the method described above not work for you do the following:Send the race(s) of your choice to your secondary profile. Log in with said profile and beat the main scores by just a few points. Now go back to your main profile and beat the scores again. Worked for many people online and some in here and hopefully for you too should you have had some difficulties.
Along the way you'll also get
and

The Contender trophy in DRIVECLUB worth 38 pointsWhen the timer expires on a Driver Challenge, be at the top of the leaderboard.Hope this helps. Enjoy your game :)
UPDATE: With the patch to version 1.11 you can be offline when the challenge timer runs out and it will still count. The Trophies should unlock next time you start the game.
Thanks to
niteowl1980,
M0j0s_D0j0,
Monkster1978,
AyyJayEmm and
Mav_Ryk_1 for contributing to this solution.