I wouldn't say that the government one day just said "Let's kill all the Indians, lol". At least in its early stages, it was more of a gradual process of white settlers gradually pushing back and driving away the local natives with or without the support of whatever government they may have had at the time.
It actually is surprisingly similar to how the early Anglo-Saxons pushed the Celts out of most of Britain until they ended up in Wales and Scotland.
However, the government naturally was anything but innocent in this - stuff like the 1830 Indian Removal act or the Long Walk of the Navajo didn't magically appear from thin air.
Consequently, I would argue that the attacks on Native Americans generally occurred in two stages : A first, early stage in which white settlers, gold miners, etc. moved into Native American lands independently from governmental support or interest, and a second, later stage, where the government actually has become aware of whatever resources the Native American land may hold and begins to pursue legal and/or military steps to drive out the Indians.
That being said, I do not think that the government really wanted to kill off all Indians (unless it happened in revenge for real or alleged previous attacks by them) - they simply wanted them out of the way, and cared little if lots of them died in the process.