The cheapest official airport parking is available in the uncovered car park P20, where you pay 40 eur per week. The most expensive parking garage right next to the terminal costs 114 eur per week. Check the airport website for details.
There are many private car parks around the airport and parkvia.com can help you compare them, usually prices for a week's parking range from €30 upwards so it's more likely to be worth leaving your car at the airport.
You can park forfree in the P+R car park at the exhibition centre (Leipziger Messe), take the "Leipzig-Mitte" exit off the motorway. There is an S-Bahn stop right next to the car park, which will take you to the airport in 5 minutes for 5,40 eur.