It takes for ever- maybe 100 hours of driving. i don't even think driving well makes a difference and it does absolutely nothing so I really wouldn't worry about it. if you're going for 100% you'll probably get it just from playing eventually.
It takes forever. :/ Finishing Driving School helps, but other than that, I don't know of any really fast ways to increase it. Ie. I don't think it has anything to do with whether you crash or not, but has to do with the number of miles you put in.
I've always had suspicions that Driving Skill did go up faster for driving "well".
I've noticed that my driving skill tends to go up after I've done something skillful like an insane jump or two-wheel bonus. Maybe I only NOTICE it more after stuff like that, but it SEEMS to be correlated, IMO.
"Kids: if you REALLY want to piss off your parents, buy real estate in an imaginary place!" -Busdriver