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The apartment we rented when we visited Paris was right across the River Seine from Notre Dame, so I had lots of opportunities to take photos of the huge church at different times of the day. There are two bridges facing the front of Notre Dame that offer great views of the church, with the river and trees in the foreground. At most times of the day there are people walking or sitting along the river, adding a bit of life to the photo.
As with most photos, it is always best to avoid the middle of the day to photograph architecture, and that's true for Notre Dame as well. One issue with Notre Dame is that the sun rises behind the building, so you will want to get ready to take photos before the sun gets too high in the sky during the morning hours. It can look cool with the sun just above the horizon, giving a bit of a halo to the church. I didn't get up quite early enough to see Notre Dame before the sun rose, but the sky can be very dramatic just after sunrise as well.
I liked taking pictures of Notre Dame with the river in the foreground because the courtyard leading up to the church isn't the most exciting. It is a fine gravel with large raised flower beds. It's not hideous, but it's a bit monotonic in my opinion, unless you are looking straight onto the building, as there is a nice brick pattern from that perspective. However, straight-on photos aren't necessarily the most exciting, so it's good to capture the church from an off-centre viewpoint. Something else I like to do is to capture only a small part of a building in the frame of the photo, and grab a piece of another object in the same frame, I like to look for nearby trees or lamps that will provide some contrast to the architecture of the building.
As with all landmarks, I really enjoyed taking pictures of Notre Dame at night. The really great thing about Notre Dame is that it sits so close to the River Seine, so you can get some great photos of Notre Dame with the water reflecting nearby lights. There are also lots of tour boats that travel down the river all day long. At night many of the boats do a dinner cruise. Although to didn't capture one from this angle, it is cool to take a photo with a boat making it's way down the river.
As with most major landmarks, the front facade is the view that gets the most attention. However, the back of Notre Dame is gorgeous. There is a small, quaint garden and fountain within the walls of the church. It's a beautiful little area, and the architecture of the back of the building was surprisingly nice, with lots of detailed stonework, and you can see the buttresses from here, which you can't see very well from the front of the church. This was a nice area to stop for a snack and a drink during an afternoon walk around the area.