Lincoln famously said, "One war at a time" - referring to the possibility of war with Britain. He had many wars at a time! What was happening to Native people in the far West and on the West Coast was of singular importance to the running of the Civil War in the east. Remember, Indians were not citizens of the United States. Most of the eastern people were diminished in power and unity - the great nations of the Creeks, Cherokees, Iroquois, etc were weakened and a minority. One fourth of Southern slaves were actually Natives. There weren't many choices, to be frank. If your remaining land and people were in the North, that's who you fought for - in the South, same thing. Slavery was an issue there - just as with the planters and for the same reasons. In the West, the issues of Indian troubles were complex. There were territories, such as Colorado, wishing to become states and ambitious men wishing to be senators or president, like Chivington. The best way then to get elected to influential political office was by killing a batch of Indians. With federal troops largely withdrawn by the war in the east, that left militias and volunteer groups to deal with Indians. Reading newspaper accounts of the 1850s-1870s tells what was going on - at least the settlers' side of it! Somebody else didn't have a voice - at all.
The Indian people who fought in the Civil War did so, as mentioned above, for just about every reason there was. Some to preserve what little was left to them, some just for the chance to kill some white men legally. (Seriously.) There were some overtures by both Confederate and Union authorities but nothing really serious. What the Confederates actually hoped to do was what they perceived the Union to be doing to them - get a minority group up in arms and causing trouble. If the Union would use their slaves against them, then let's get the Union's Indians to rise up against them. Turnabout is fair play! Nobody was considering that both the slaves and the Indians had their own reasons for doing what they did. The problem with this plan the Confederates had was it couldn't work with Western Indians. They didn't give a durn about any of that - invaders were invaders whether they were Union, Confederate, Mexican, French, or anything else. One of the effects of the Civil War in the newly admitted state of California was the worst destruction of Native people since Cortez hit Mexico. From about 1850 to 1870, the Indian population of California was reduced by 90 percent. California Indians didn't know about, or give a hoot if they did know, about the American CW. But it withdrew what little protection they did have by withdrawing federal troops. Not that they were innocent of atrocity, they weren't, but the miners and ranchers had a field day with nothing to stop them. That left the states to deal with them. California put bounties out and raised militias that way.
The 19th century really wasn't the greatest time to be a Native!