Spring is best for wildflowers and late fall is usually manageable, temperature-wise.
Don't expect much rain. With an average rainfall of a measly 1.75 inches per year, Death Valley National Park is one of the driest places in the country. Best months to make the ascent are May, June, and October. During the winter, you'll need to be prepared for snow and ice climbing at higher elevations. Be sure to register at the visitor center's backcountry desk before heading out.