It's about a 6 hour drive from Ottawa to Pinkham Notch. Allow for extra time if you hit Montreal during rush hour. You might consider travelling only as far as Twin Mountain or Gorham and stay at one of the many inexpensive motels, especially if you're leaving late in the evening. Alternatively, you may wish to stay at the Joe Dodge Center, which is located at the Pinkham Notch trailhead (book ahead).
Take the 417 east to Montreal, follow the 10 east on the south shore until the junction for 55 south. Follow 55 south to the US border and the state of Vermont. Head down Interstate 91 south to St. Johnsbury and east on Interstate 93 crossing into New Hampshire. Follow I93 to Bethlehem and continue east on 302 until Twin Mountain. Continue east on 115 which then meets route 2, eventually delivering you to Gorham. Follow the 16 south to Pinkham Notch.
Montreal is a 4 hour drive. Same route as from Ottawa.