Solution from Uncle Manwich user.
You climb the ranks by earning Kudos which will earn you the Rank 10,5 and Rank 1 achievement.
Rank 10 - 1,000 Kudos
Rank 5 - 160,000 Kudos
Rank 1 - 475,000 Kudos
There are multiple ways of getting this.
-Complete offline events (Gotham Career). The harder the difficulty, the more Kudos you earn after each race. Try linking drifts together, avoiding collisions and remember to draft your opponents, you get Kudos for winning in style. Remember that you MUST finish the races for your Kudos to be counted.
-Complete online events (Online Career). This method is harder than offline races since it is easier to lose and you get roughly the same amount of Kudos.
-If you are used to playing racing games, try the cone challenge events on the Hardcore (Platinum) difficulty. Avoid collisions, keep your combo and string the cone gates with drifts! Using the Ultima GTR, it possible to make 10,000 to 50,000 Kudos per event.
Just keep improving your skills and keep playing!