The general outline contained in this guide also applies (obviously) to the 25%, 50% and 75% of career achievements this game contains.
The first thing to say is that you can't actually NOT get this achievement. The reason I say that is because, at any time, you can stop and re-start a race within a championship and not be penalised.
If you have done two events of a three event career challenge, but you've had a bad start in the third event, simply pause and re-start. You can continue to do this for as long as it takes for you to get the result you desire.
Some general tips that have helped me through 24 events with only a minimum of re-starts:
1 - Race on 1000cc. Simply put, your car is easier to control, you have slightly more time to avoid problems, you can nail KERS corners better, slipstream more efficiently, take shortcuts more easily. Yes the races are longer, but you won't have to re-do as many of them.
2 - Pick a favoured driver/team and stick with them. There is an argument that some teams and their power-up abilities are best for certain events, but I have used the same team (Red Bull) for all events and have had few problems.
The Red Bull power-up - slipstream boost - I have found invaluable for a variety of events. Races obviously, eliminations, sector snatch, points/pole position, even slalom. I immediately slipstream as the race starts, get a boost, use it, slipstream, get another boost, use it, etc. It is invaluable in getting your car through the field early, out of the pack and out of the vast majority of trouble.
3 - Linked to point 2 - Your aim in this game, and in career, must be to GET OUT INTO THE OPEN and race cleanly. Being able to get some clean running is vital.
4 - Don't feel compelled to win every race. Race cleanly, quickly and know when to attack for a spot further up the finishing order (if your car is in good condition, you can slipstream or use boosts to catch, etc) or when to settle for second or third place (damaged car, others too far in front, people behind attacking you - meaning you must defend with power ups).
You don't have to win every race within a career event to get the 3 stars ... a consistent effort is preferable to winning all the time.
5 - Know your shortcuts. The AI cheats and drives brutally on this game. Knowing your shortcuts and using them is vital to level the playing field and keep you in the game.
6 - Be merciless - Ultimately, the AI will attack you and attack you and attack you and damage you. You may as well do the same back.
7 - Pitting - Pick and choose your spots. If you can pit and not lose too much ground (or if cars around you are all pitting) do it. If pitting will slow you down too much, maybe wait and get to the next pit stop. You'll have to make these decisions on the fly in the race.
8 - Know the race mode but get to the front - Each race mode has a different strategy, but the vast majority require you to get to the front and stay near it. Get to the head of the pack early, get some clear driving space and give yourself the best chance of taking sectors, racking up points, avoiding elimination or simply winning.
9 - Pay attention to the race restrictions/conditions - Very important. Some races are pure skill driving with no boosts. Some have double damage. Some have fragile cars. Adjust your driving accordingly.
(As a point - the "fragile car" races are THE WORST. The AI drives like kamikaze pilots and any jostling with them will damage your car badly. But they'll do it anyway and not care about the damage. One race in particular - in Germany with the tight left hand shortcut after the first KERS corner - is nightmarish. It is the one race I restarted ... and I restarted it a dozen times in order to get a clear run to the first corner. Ultimately I finished 3rd and was happy with that!)
10 - Patience. Don't hurl your controller at the cheating AI. Take a break, pause, restart, grab a drink, refocus and have another go. Seriously, once you are in the groove, it isn't too bad, and beating up on the AI can be seriously satisfying.
Final point - jump boost as often as possible. Crucial for getting a little extra speed within races.
Best of luck.