Tempranillo is a grapevine (Vitis vinifera L.) variety extensively used for world wine production which is expected to be affected by environmental parameters modified by ongoing global climate changes, i.e., increases in average air temperature and rise of atmospheric CO2 levels. Apart from determining their effects on grape development and biochemical characteristics, this paper considers the intravarietal diversity of the cultivar Tempranillo as a tool to develop future adaptive strategies to face the impact of climate change on grapevine. Fruit-bearing cuttings of five clones (RJ43, CL306, T3, VN31, and 1084) were grown in temperature gradient greenhouses (TGGs), from fruit set to maturity, under two temperature regimes (ambient temperature vs. ambient temperature plus 4°C) and two CO2 levels (ambient, ca. 400 ppm, vs. elevated, 700 ppm). Treatments were applied separately or in combination. The analyses carried out included berry phenological development, the evolution in the concentration of must compounds (organic acids, sugars, and amino acids), and total skin anthocyanins. Elevated temperature hastened berry ripening, sugar accumulation, and malic acid breakdown, especially when combined with high CO2. Climate change conditions reduced the amino acid content 2 weeks after mid-veraison and seemed to delay amino acidic maturity. Elevated CO2 reduced the decoupling effect of temperature on the anthocyanin to sugar ratio. The impact of these factors, taken individually or combined, was dependent on the clone analyzed, thus indicating certain intravarietal variability in the response of Tempranillo to these climate change-related factors.