Going from LA to DVNP and then to Mammoth Lakes in one day is going to be packing an awful lot into that day. LA directly to Mammoth Lakes is 4.5 hours or so by itself, which wouldn't leave much time for a side excursion to check out DVNP if you want to get up to Mammoth Lakes by the evening. As Otis mentioned, you could maybe shoot into the west entrance of DVNP by taking Highway 190 east from the Olancha area and see Crowley Point, which is a great view. You might have enough time to do the hike to Darwin Falls. But that's about it. I wouldn't recommend going down into the valley proper in late June -- it's going to be hot down on the valley floor. Stay at higher elevations where it's cooler. As Otis said, take plenty of water -- more than you think you'll need.
FWIW, if you're a U2 fan, the site of the joshua tree that was on the cover of their album The Joshua Tree is located a short hike off Highway 190 not far outside of DVNP. If you google "location of U2 joshua tree," all kinds of websites will pop up that can provide the exact location, including GPS coordinates. The tree itself died several years ago and is now a carcass on the ground, but the site still sees a lot of pilgrimages by U2 fans and music fans in general. They leave all kinds of memorabilia and other shit out there, including a heavy-ass plaque. The wife (no pics) and I found it one time for kicks when we were coming back from a trip up to Mammoth and we had some time on our hands. It's like a 5-10 minute hike from the side of the highway if you know where you're going. It's an interesting little side-side-trip that probably wouldn't eat up more than 30 minutes or an hour at most. But, again, it's very likely to be quite hot in that area during the day in late June, so make sure you know exactly where the location is when you park your car and start walking, take water with you, wear comfortable shoes, etc., even though it's only a short hike from the highway to the site of the tree.
If you get dissuaded from doing the side trip into DVNP, you still have some other options. The Alabama Hills just outside of Lone Pine are very cool -- the site of many movies over the years. Easily drivable (the roads are soft dirt) and some short hikes make it fun for kids . . . assuming you'll have kids in tow. Kids love climbing on the rocks. You can spend anywhere from an hour to half a day in the Hills. There's also a pretty good little film museum in Lone Pine that focuses on all the films that have been made through the years in the Lone Pine area.
Or stop at the Manzanar National Historic Site right off Highway 395 between Lone Pine and Independence. Well worth a few hours or even a half day. Terrific visitor's center with a lot of informative exhibits, movies, etc., and you can also drive around the area that formerly served as the intern camp, get out and walk to where buildings once stood, etc. There's also an obelisk-like memorial. Visiting Manzanar is going to be very interesting (and sobering) for anyone with even a passing interest in WW2 or history in general. Frankly, I think you'd probably be better off checking out Manzanar instead of doing the quick side trip into DVNP simply because (as Otis mentioned) most of the cool things to see in DVNP are either located in the eastern side of the park in the Furnace Creek area which will take extra time getting to and where it will likely be hot as hell, or are closed for repairs (Scotty's Castle). Save DVNP for some other time (esp. the spring, when the flowers are blooming and the temps are comfortable) when you have more time to explore it.