I'll give my two cents on this campaign. Like with the Confederacy, the California republic cannot get supplies and materials passed the Union blockade. They are able to smuggle materials through the borders and wilderness areas between Canada and Mexico to bring in many alternative supplies through contraband. Because of agriculture in the state, California generally becomes self-sufficient in food production which can nullify much of the effects of the American blockade. It does cause some damage over the economy too however.
Californians will fight a different type of war to maintain their independence. Soldiers from California first help the Confederacy in its attempts to establish control over New Mexico and Arizona. This creates supply lines between Texas, the Confederacy, and California in order to support the war effort. Nevada never becomes a state because Californian troops are able to secure its main points to use much of the resources for self-defense against the Union.
The first years of the war prove not very effective against the rebellious states. Union soldiers from other territories like Colorado are sent to suppress the Californian invasion to great defeat. Californians are able to do something not as possible in the South, use the terrain for the purposes of guerrilla warfare. Union militias are stopped this way. It's not until Union forces under Grant are able to seize control over the Mississippi River after the Siege of Vicksburg and cut off supply lines to California and Texas does the situation change. Union volunteers are able to muster up a force from some of the Midwestern states to first invade Texas and claim its major cities. Confederate forces withdrawal from New Mexico and Arizona in and effort to defend Texas from incursion. Eventually this fails by the beginning of 1864. California troops are also used in this matter. Eventually a smaller army under Union control enters Nevada through Arizona and Colorado to reclaim territory. Californian resistance retreats to its state. By the middle of 1864, California is back under the control of the Union and the rebellion is crushed there. However, guerrilla resistance, especially in the mountains remains as an obstacle until the end of the war.