The most accessible route is from the end of the publicly-accessible paved road in Jawbone Canyon.
This is an easy class 1-2 (more like class 1.1) route; it is about 7 miles long with 2,500 ft of elevation gain. Routefinding is trivial, but you do need to know where you are going at all times - keep your wits about. Depending on your fitness level and experience with cross-country terrain, the hike can take anywhere from 3 to 8 hours. The average SP reader will probably spend 4-5 hours on the mountain.
From wherever you are, you need to get to Jawbone Canyon. From California State Route 14, turn west onto the paved Jawbone Canyon Road. If you are coming from the south (Los Angeles), this is a left turn; from the north (San Francisco/Bishop), this is a right turn.
Immediately pass the visitor's center on your right. You'll go over the big pipe of the Los Angeles Aqueduct twice. After about 6 miles from the highway, you'll reach a gate on the paved road. Just before this gate is a parking area on the left; park here.
Hop the gate in front of you. There is no trespass issue; this is all BLM land. If you get "caught" by area staff, they will ask you what you are doing and where you are going. Mention you are hiking and trying to bag Cross Mountain, and they will leave you alone. The point of the gate is to restrict public access to the wind turbines visible in the distance.
Option 1: Follow the paved road about 1 mile to a seasonal cattle corral. On the topo map, this is where the Jawbone Well is. This is the mouth of a very wide side canyon that you want to follow for a short distance; you may soon pick up a semi-distinct use trail. From the canyon, you'll see that there is a low ridge to the right that you want to get onto. As you go into the canyon, you'll pass several "finger" ridges extending down from this ridge into the canyon. Climb one of the fingers; the easiest one to ascend is the finger closest to the mouth of the canyon. However, the ridge undulates up and down, so you may save some time if you pick one of the fingers further up the canyon. Be careful not to go too far into the canyon; the place on the topo map marked "Spring" is about as far up the canyon as you can get before you lose the route.
Once on the ridge, follow it uphill to the base of a giant scree slope. There are some motorcycle tracks and a big firebreak that will get you close to this slope. Slog up the scree slope to a saddle. From the saddle, turn west and follow slippery use trails to the summit.
Option 2: Alternatively, you can get onto the low ridge straight from the paved road. Go past the corral, and go up the first ridge you see on the left. This will take about 5-10 minutes longer compared to Option 1, but this makes the route somewhat simpler. Once on the ridge, continue as indicated in the second paragraph of Option 1.
Option 3: On the paved road, pass the corral and the base of the low ridge described in the other options. Right after the ridge is a canyon on your left with a dirt road. Follow this road to near Cutterbank Spring. Once at Cutterbank Spring, you are on the low ridge. Continue as described in the second paragraph of Option 1.
To descend, go the way you came up.
Bring your Ten Essentials and good hiking boots. Poles help with the scree slope, but are not necessary.