This is the hardest achievement in the game. Let me break down for you what the expert difficulty entails.
*All assists off--This includes Traction Control, Automatic Gears, ABS, Racing Line, and Brake Assist (which is useless, so no big deal losing that).
*Simulation options set to full--This includes realistic rules and flags, tyre simulation, fuel simulation and full damage.
*The AI is set to legendary difficulty.
The biggest hurdle for most will be the lack of traction control and the use of manual gears. Without traction control, spinning becomes a common occurrence. Even with the slightest mistake, you will be punished. My advice, never brake when turning. EVER. It's just asking for trouble. Go wide, brake, then turn. Do not push full throttle when coming out of a corner. Ease back up to speed, or you will spin. Be very wary of the kerbs and do not go off road.
Learn the tracks in and out. You need to know the best lines. Also, opt for consistency. No point going flat out if you're going to spin and lose more positions than you gained. Your car setup is important too. Check out this thread for more hints.
The achievement must be done in Career, so you are going to need to wait until you get contract offers from better teams, to have any chance of winning the Championship on expert.
At the end of the day, nothing beats practice.
Alternative Method (Easier)
Start the season (first race, Bahrain) on expert. When you get on the track, enter the pause menu and change the assists to whatever is easiest for you. You're going to trick the game into thinking that you are playing on expert. Before ending the session, whether it be practice, qualifying, or the race, enter the pause menu again and set the assists back to what they originally were (outlined above). The game autosaves at the start and end of the session, so you can trick it in the middle. Continue to do this for the entire season, win, and the achievement will come. A few things to note:
Do not save a setup whilst you have any assists on. This will cause the game to autosave, thus ruining the exploit.
You can restart a race with assists set to on, same for qualifying.
You can enter the garage with assists set to on.
As long as the game doesn't Autosave when you have assists on, the exploit will work.