Your choice of where to bicycle through wine country may be dictated by the constraints of budget and time.
If you're in California, you might want to head for Napa or Sonoma. The next consideration is the season. By mid-spring, you can expect mild temperatures and clear days. Spring is a great time to see wildflowers, too. Summer will be warmer, and the grapes will be ripening. Fall can be hot, but you can still see the grapes get picked and crushed. If you choose to go during the winter, take rain gear.
In Virginia or North Carolina, you can tour the rolling hills of the Piedmont region and the foothills of the Blue Ridge Mountains. The dead of summer is likely to be too hot and humid to be pleasant, and winter can be cold and snowy. Late spring, when flowers bloom and the weather is pleasant, or fall, when temperatures get more moderate and leaves blaze with color, are good times for biking in wine country.
If you can go further afield, there are other considerations. A major one is the kinds of wine you're interested in. Maybe cycling through any wine country will be fine. But if you're a serious oenophile (wine lover), you may want to research wine regions. If you love Sauvignon Blanc, for example, head for the Marlborough region of New Zealand's South Island. Crazy about Malbec? You can take a bicycle tour through Argentina's Mendoza Province. Tour companies will provide advice on the best time of year to ride.