Hi, and welcome to JA!
That's a great question.
White soldiers made about $13 per week, and black soldiers made about $10 per week (http://library.thinkquest.org/CR0212661/id40.htm).
There was no direct deposit back then. This picture should give you a pretty good idea of what the soliders did to collect their earnings:
This link from the National Park Service explains that some soldiers mailed their earnings home to their families: http://www.nps.gov/archive/gett/soldierlife/cwarmy.htm
In doing this research for you, I spoke with National Park Service Ranger Frank at the Fredricksburg National Military Park in Virginia (http://www.nps.gov/frsp, he can be reached at: 1(NNN) NNN-NNNN. He explained that the payments were in cash, and that they did not happen regularly (even though they were supposed to). A quartermaster would visit each set of troops about every 2-6 months to give them their cash payments. I also spoke with a Ranger from Appomattox Court House National Historical Park, also in Virginia (http://www.nps.gov/apco/). That ranger explained that the Union troops were paid by the Federal Government, while the Confederate troops were paid by the individual states that they were from.
I hope this helps. Let me know if you need any more information.
You're welcome! Thanks for the accept, bonus, and feedback!