If you choose to go backpacking in the Sierra Nevada in April, prepare yourself as if you were going in winter, because there will still be significant snow, and you could be hit by a blizzard at any time.
To quote the National Park Service:
Hiking and backpacking options are still limited in April. Expect snow above 6,000 feet (this varies from year to year). The Mist Trail to the top of Vernal Fall usually opens sometime in April, although it is possible to reach the top of Vernal Fall via the John Muir Trail and continue on to Nevada Fall. The Yosemite Falls Trail is typically snow-free at least most of the way to the top (but snow is likely at the top, making for a snow-camping experience if you're backpacking). The Four Mile Trail to Glacier Point is closed (but may be open to Union Point, about halfway up).
Outside of Yosemite Valley, snow may still be present in the Upper Mariposa Grove, but the lower grove is usually mostly snow-free. Chilnualna Falls Trail (Wawona) may be mostly snow-free and hiking to Wapama and Rancheria Falls (Hetch Hetchy) is another usually snow-free option.
There are no multi-night backpacking options in April that do not involve hiking and camping in the snow