Flying horizontally with the camera pointing out the side is also a bad idea for the aforementioned reasons, but also, rockets will spin in flight due to the slightest misalignment of the fins. A rocket pointing a lens out the side flying horizontally will shoot the ground only part of the time.
If you want to build a rocket, get a kit. Its cheap and it has most of the stuff you'll need like the engine mounting hardware, a parachute, fin material, and the nose cone (difficult to make by hand). Many are designed to be payload carriers as well.
If you go the route I took, that is, tearing down an old movie camera, the rocket you fly will have to be the "high power" variety due to the camera's size. Mine was about 4-5ft tall with a 4inch diamter body. There is too much info around high power rocketry to cover here. Check out:
Tripoli Rocketry Association
As for my camera, here are some pics of two I made.
This was the first. There was a lot of metal in it, so I drilled it full of holes to lighten it. The orange triangle piece is a mirror housing I cut off from a toy periscope. Since the camera lens sat pointing up in the payload section, the mirror angled the view horizontally. Another mirror on the outside of the rocket in an aerodynamic shroud angled the view again back towards the ground.
Here's the second camera, mostly plastic construction so I left a lot intact.
Power for both came from a 9v battery. I'll see if I can dig up some pics of the actual rockets I used. They're long gone now (gave them to a budding rocketeer).