$5k - $10k definetly not an good budget with racing games, but its still decent budget to acquire installs + revenue.
First Month:
1. Spend atleast $500 for aso, recruit decent aso company who know what they are doing. This should be done before launch. Search in google and you will find good aso professionals.
2. Don't do any promotions (buying installs) other than aso, buying reviews, press release. Aso process lasts for one month because aso firm should provide you second set of kws, new title and desc after 14 days. So in a month you will be updating kws accordingly.
3. submit your app to app review sites, 70% of the sites will not post your app, but still give it a try.
4. Spamming the gaming forums. yes, hire some guys from DP and ask them to post in major gaming/app forums about your app.
Second Month:
1. Continue your aso. (perfect aso takes maximum 3 months to find perfect kws and title.)
2. Start buying installs from FB, as it is cheap. Combining with your kws, your app will be boosted well). If your budget is $2000 for app installs, dont spend $100 per day, do it all at once. More installs in a day will move your app to Top 10 in your game category. When you hit top 10, you get natural installs, where your app remains there.
3. Buy reviews and ratings from micro workers.
If your game is multiplayer version, add some incentives like $25 amazon gift card per week for who achieves leader board points. This brings you more IAP revenue.
If you can pm me your game in detail, i can suggest you few more things.
Good luck.