At the beginning when that scene takes place he is using his personal firearms, which are a combination of Pennsylvania Rifles and shotguns. The former are quite accurate out to at least 100 yards. Later on they are using Brown Bess muskets which were standard issue for the Continental Army.
Better than that. Timothy Murphy, one of the better than average shots, was reported to have qualified for Morgan's Rifle Corps with repeated hits on a seven-inch target at 250 yards. But Murphy's skill as a woodsrunner and his ruthlessness at dispatching a good target was certainly at least as much a factor as his marksmanship; and it was Murphy credited with eliminating both Sir Francis Clark and General Simon Fraser at the Battle of Saratoga, leaving the British leaderless and confused until adjutant's could decide which would take over.
That was less the case on the other side, at least at times. British rifleman and ordnance expert Major Patrick Furguson was probably at least as good a shot as Murphy, but when at the Battle of Brandywine a Colonial officer presented himself just in range, Major Furguson could not bring himself to shoot the man in the back... and the man who may have been George Washington went on to accomplish other things.