The Portsmouth Trojans have won 14 league/conference championships in school history. Here is a list of those titles, with overall record included:
1926: 13-4..........Southeastern Ohio Athletic League Champions
1928: 14-3..........Southeastern Ohio Athletic League Champions
1939: 15-7..........Southern Ohio League Champions
1955: 20-4..........Greater Ohio League Champions
1961: 22-3..........Greater Ohio League Champions
1979: 21-1..........Tri-State Athletic Conference Champions
1987: 22-3..........Ohio-Kentucky Athletic Conference Champions
1988: 27-1..........Ohio-Kentucky Athletic Conference Champions
1989: 21-4..........Ohio-Kentucky Athletic Conference Champions
1990: 24-3..........Ohio-Kentucky Athletic Conference Champions
1991: 22-3..........Ohio-Kentucky Athletic Conference Champions
2012: 24-3..........Southeastern Ohio Athletic League Champions
2014: 19-5..........Southeastern Ohio Athletic League Champions
2015: 20-7..........Ohio Valley Conference Champions
Update June 9, 2014:
I had originally posted that the Trojans won the Greater Ohio League in 1965, but upon review of their schedule in the 1965 yearbook, PHS won at Middletown 63-62, but played no other GOL schools. Evidently the league was being restructured at the time or had temporarily disbanded, because in the 1970's the Trojans were playing Lima, Springfield South, and Middletown again. The league did disband for good by the mid 1970's.
Updated October 14, 2014: I discovered that Portsmouth tied Middletown for the Southern Ohio League championship in the 1938-39 season. Portsmouth, Middletown, Hamilton, and Springfield were members of the league, and the SOL was probably the forerunner of the Greater Ohio League which would start several years later, with Lima, Mansfield, and Chillicothe as members at some point in the league's lifespan.
Updated October 14, 2014: The Trojans also won SEOAL titles in 1926 and 1928, and then left the league. Nelsonville won the league title in 1927, however, Portsmouth advanced to its first Final 4 that year.