***UPDATE***
Players are reporting that there's a (good) bug where if you're at least level 50 then you just have to play one more championship in the Hot Wheels Horizon Open to unlock the achievement. I tried playing a Hot Wheels championship after already having level 100 and it gave me the level 200 badge. My guess is that switching between the regular Horizon Open and Hot Wheels Horizon Open (or vice versa) doubles your current level temporarily. So if you're between levels 51-99 while doing regular Horizon Open then you can try to do a Hot Wheels Open championship and see if the achievement unlocks early for you.
***END UPDATE***
The quickest way I've found to level up is to do S2 Street racing.
1. Select a fast, easy to control S2 class car. My suggestion: Koenigsegg CCGT with tune 732 604 811. Another option from the AutoShow is the Ferrari FXX K with tune 955 500 683. If that car is too expensive then any S2 class car is fine. The vast majority of XP will come from simply finishing and not from performing well.
2. Go to the Online tab of the menu and select Horizon Open.
3. Select the Custom Racing option towards the right.
4. Set Racing Series to "STREET" and Performance Class to "S2".
Street races will only be point-to-point so you won't have to deal with long, 3-lap races. Plus, being S2 will make the races end really quickly. You only accumulate Horizon Open XP at the end of the championship so be sure not to quit until the 3 event championship is finished.
This post from the Forza forums claims that it takes 4k XP to go up a level from levels 0-100. You get 20k XP(5 levels) just for finishing a championship. Plus another 5k XP if you win. This matches up with my experience. I would level up about 6-8 levels after every championship with this method.
If we say you gain 6 levels after every tournament and an average tournament takes 15 minutes then that would take you about 4 hours to go from level 0 to level 100.
Per MysterioBadass, also check your car's mastery for any XP boosts. For example, the CCGT has a mastery you can purchase that permanently boosts your event finish XP by 20% which can allow you to level up noticeably faster.