To reach level 10 online, you need to earn somewhere between 240,000 - 290,000 credits in multiplayer events, depending on the player, events and impact rating. You can also earn credits in the global challenge at the end of every week, or custom/private games but it is much lower than online races.
The easiest playlist to level up in is the "Alternative" playlist where you will take part in Endurance races, Time Attack, Drift, Touge & Face-Off events. Credits are determined based on your finishing position, as well as your driver rating. Clean drivers will earn a substantial amount more than dirty drivers. Winning in a touge or face-off event should net you between 4000-4500, and playing well in full 12-player event such as Drift, Endurance race or Time Attack can gain you from 10,000-30,000 credits.
Remember to level up your cars to gain a small advantage over others. This achievement should take between 5 to 8 hours for the average player, depending on their skill.
Fast and easy boosting method (courtesy of Creech):
Go to GRID Online, choose Events, Xbox Live, Online Custom Event, Create Game. Change Race Type to Touge, Rounds to 5. Get one boosting partner and trade wins each round. Note that if the losing player in a given round just sits at the starting line, the game will kick them out. They need to just progress slowly down the track at 30mph or so until the winning player is 5 seconds ahead of them.
Each person will get just under $10,000 every 7 minutes or so.