I am writing this with my past experience and not as an expert

Private car should be owned by you.If it is under bank loan you have to get a clearance letter from the bank. umrah visa will be processed by many travel agencies.
When you enter the Saudi boarder you should take saudi insurance for the car.

Doha to Makkah it is almost 1500 km and you can go thru the saudi national highway No 40 thru Hufuf, Riyadh,Taif
Near the Thaif is the place for entering Ihram.(meekath)

Makkah to Madeena almost 450 km

From Madeena to Riyadh ther is National Highway which passes thru Buraidha and from Riyadh to Doha the same highway described before

All these high ways you get get petrol bunk,shopping markets, vehicle repairing garages and resting places almost within 1 to 2 hours journey interwals

I hope this will help you somewhat