Unless you're a real life F1 driver, I suggest you stick with hard rather than expert. Here is what the hard difficulty entails:
Traction Control on Medium.
Full Simulation Settings (Tyre wear, fuel etc).
No Dynamic Racing Line.
The pro AI are no slouches, so you're going to need to be at one of the best teams, such as Red Bull, Ferrari or McLaren. Therefore, you won't be able to attempt this until at least your second season in Career.
The biggest obstacle for most will be traction control, followed by manual gearing. With traction control on medium, you have a little bit of lee-way with your throttle application, but it is still easy to spin. Basically when taking corners, you want to go around them at a constant speed (relative to the corner). When exiting, if you apply too much throttle your rear tyres will spin, losing traction. This usually ends badly. It takes a lot of practice to get used to driving without full traction control. On full you're basically super glued to the track. Going to medium, it's akin to driving on an ice rink.
Sudden throttle application leads to wheel spin, which leads to loss of traction. Keep this in mind when racing.
Again, when it comes to manual gears, it is all about practice. Some corners you'll be going down to 1st gear or 2nd gear, you need to test it yourself, so take advantage of the practice sessions. Shifting up is quite simple, listen to the engine and watch the rev limiter, when it's in the red or the engine starts "screaming" at you, it's time to shift.
In the races, be smart. Do not use all of your KERS on the very first straight, you may need to defend against AI later in the lap and KERS is a very effective means of defence. Stay out of trouble, be consistent and hopefully you'll come away with some solid points. You do not need to win every race!
If you want to be a little bit sneaky, it is possible to cause other cars to retire, by making them run into the back of you. For some reason, the back of your car is extremely strong. If you find yourself in front of someone, brake a little harder than usual. If they run up the back of you, they will likely retire. This is an easy way of knocking your rivals out, if they're getting too close to you in the Championship. It's not really in the spirit of the sport, but in terms of getting the achievement the easiest way possible, it is fair game.
Finish on top of the Drivers World Championship table with the difficulty set to hard to unlock this achievement.
Note: Easier method:
When you are starting your career, set the difficulty to hard before each race. Once you're on the grid, enter the pause menu and change the assists to whatever you are comfortable with. For example, you can change manual transmission to automatic, traction control to full etc. You will still be racing professional AI and there will be tyre wear + fuel to take into account. When you are about to cross the finish line, enter the pause menu and reset the assists back to what they were on hard. This tricks the game into thinking you have been playing on hard the whole time.
You can leave auto save on (ignore the fact that it autosaves when you change assists, it doesn't matter). However you may prefer to turn it off and save manually, as it is sometimes easy to forget to change the settings back to hard.
Make sure you don't save any setups while you have the settings changed. You can restart a race with the assists changed, just make sure you change them back to hard before ending the session.