Various states kept their records differently. With the 57th Ohio (the unit I generally work on), I start with the Official Roster of the Soldiers of the State of Ohio. I'm not as familiar with the 106th New York, but you would probably start with the 106th Infantry Roster found linked at the bottom of the space page in the original post above.
It tells us he was 43 when he enlisted in 1864, so a birth date about 1821. His enlistment date, his company, and his discharge date.
Using that information, I move on to Fold3, looking for pension records. They will often have a date of death (for when the pension stops) and they will also often have a widow listed. For New York regiments, it appears Fold3 also has Muster Roll Abstracts, which repeats exactly the information found on that 106th Infantry Roster.
In his case, neither a date of death or a widow appears on his pension documents. This seems to be because he was denied -- this may be because he died before it was approved, or he got into trouble somehow. That is not clear, as the application itself is not available on Fold3 at this time, only the index card. However, it did list a variety of units he also served in -- the 106th NY, the 77th PA (Co G and H), the 159th Company, 2nd Battalion Veteran Reserve Corps, and service in the Mexican War. It also included the application date, which tells us he was still living in 1892.
One step better, I also managed to find on Fold3 a request to the Adjutant General in 1845 from his mother, Phebe, for his discharge. It lists his unit, where she was writing from, that her husband was deceased, that he had no living siblings, and that he had joined two years previously in June.
With that information, I could confirm the information found in the register of enlistments (also on Fold3).
Using all that information from Fold3, I then move on to FamilySearch and Find a Grave.
His grave (found on Find a Grave) aligns with the 77th PA, as Andrew posted. The location of the grave makes it pretty clear it's the same man appearing in Luzerne County, PA, on census documents. His mother, Phebe, appearing on the 1850 and 1860 Census confirms it.
A little further searching on FamilySearch turned up his death record in Pennsylvania.
After that, I usually search on Newspapers.com for an obituary around his date of death and location. I was unable to dig anything up for him. For people local to my own location, I know newspaper archives are available through the local library online, and will search there as well. I did not look for a Luzerne County Library which may have something similar.
Hope that helps. Obviously, if you don't have access to Fold3, this process won't really work.
