In S01E08- a letter was sent from Grand Maester Pycelle to the Lord Commander Mormont (the Old Bear) declaring that Eddard Stark has been declared a traitor to the kingdom (of Joffrey Baratheon).
Mormont chooses to share the letter with Jon Snow, so at the very least Jon knows that his father had been declared a traitor for not 'bending the knee' to Joffrey when Robert died. From the book:
"Lord Eddard Stark has been imprisoned. He is charged with treason. It is said he plotted with Robert's brothers to deny the throne to Prince Joffrey"
Its also hardly a secret that Ned favored Stannis to be the next King, soon after Robert dies, Joffrey calls everyone significant present in Kings Landing to the throne room to receive oaths of fealty from them. Ned had hoped that Cersei would flee the city with Joffrey, but he was forced to denounce him in a public way. From the book he declares:
"Your son has no claim on the throne he sits. Lord Stannis is Robert's true heir.
This declaration was made in front of all the significant people of court at King's Landing.
When Ned is taken prisoner the whole kingdom at this point descends into civil war, with the Lannister, Stark and two distinct Baratheon armies (Stannis and Renly), plus all their banner-men fighting over control of the kingdom.
It seems very unlikely that the news that Ned had declared himself for Stannis was not heard by Jon at some point.