There are plenty of written letters from troops during the Civil War - Ken Burns' documentary on the subject frequently cites them. They also display an incredible variance in the degree of literacy from soldier to soldier. The same Union that had a trooper writing like this:

Ituk this time tunnite to informe you that iwas well and hope these fieu lines will finde you all well. We are in roller County. Yesteday we went 40 milse back to flat top mountain. If we would stade on the easte river twenty four ours longer we woode aseede Richmonde (1)

Also had a soldier writing like this:

Hearing that there is a mail going out this afternoon I thought I would improve the opportunity and write you a few lines.

I am sorry to tell you I am not very well at present. I was taken sick about three weeks ago with chronic diarhea, and have been in the hospital about a fortnight. I have been pretty sick but I am getting better now and hope to be well in a few days. (2)

Regional variance aside, literacy would also depend tremendously on where immigrant Americans came from and how long ago they made the trek. The Nordic countries in the 19th century had a fine tradition of literacy and even peasants were literate; those under the Romanov yoke were far less likely to be educated.

Postal services were also an essential part of state services throughout the Civil War. In total, there were 28,586 post offices in Civil War America, of which 8,535 were operated by the Confederacy during the War (3).

Literacy was relatively common throughout the Union, albeit in wildly different degrees of development and depending on a host of factors. Postage was a chronic problem for the Confederacy, as were paper shortages; although services were hardly great, the importance of mail in the Civil War is worth noting. I hope someone comes by with more detailed evidence of American literacy specifically and I hope the information I brought along is helpful.