According to Rasmussen Reports, it's currently Kilgore 44%, Kaine 36%, "other" 5%, and the remaning 15% apparently undecided. That's not great news for Kaine, but it's not terrible either. The bottom line here is that nobody's got a clear lead, expecially this early in the race, and also with a "plus or minus" 4.5 percentage points margin of sampling error.
For now, this is most likely more a reflection of the state's Republican-leaning partisan breakdown than anything. We've got a looooong campaign ahead of us, including debates if Jerry Kilgore will ever agree to appear on the same stage is Tim Kaine. And Tim Kaine has been raising more money than Jerry Kilgore, which could make a big, BIG difference down the road. Anyway, stay tuned, this should be interesting!