Qassem al-Abbassi, 17, was killed by Israeli forces on Thursday night while he was in a private vehicle with his friends who were attempting to pass the checkpoint.
Abbassi is from the town of Silwan in occupied East Jerusalem. His body has been withheld by the Israeli army.
On Friday, Abbassi's family contested claims by the Israeli military that he had attempted to ram his car into the soldiers.
In a press statement, family elder Moussa al-Abbasi said that what happened to Abbassi is murder, and demanded a full investigation into the shooting.
The family also called for an autopsy to be carried out on, and for Abbassi's body to be returned to them in order to conduct a proper burial and funeral.
Maan, a local Palestinian news agency, reported that Mohammed al-Abbassi, a relative of the slain teen, was with him in the car at the time of the shooting.
"We were on our way to Nablus when we reached the checkpoint," Mohammed said.
He said an Israeli policeman approached to tell them the road was closed.
"As we attempted to go back to the main road, we were chased by either Israeli soldiers or settlers," he continued. "We could barely see as there were not enough lights and it was very dark."
It was at that point, Mohammad said, they came under a heavy barrage of gunfire from all directions.