Black will help it warm in the daytime. But it will also help it cool at night. Not what you're after (unless you're hoping that it will thaw from sunlight if it ever happens to freeze up).
Supporting it in the air lets the cold air circulate around it and pull heat from it.
Putting it underground, even a little, insulates it. (Especially if it's above the water table.)
Ideally you'd put it below the frost line (assuming you're not in a permafrost area where the freezing is BELOW the frost line). Then it probably wouldn't freeze unless you fed it slush and let it clog. But given the insulating properties of plastic pipe, keeping the freezing winds from circulating around it, feeding it with water warm enough to flow, and letting it run all the time will probably keep it running all winter.
Tap the water as low as practical.
The reason water runs under ice is that it has a maximum density at about 4C (a tad over 39F). So when it's freezing cold the warmer water sinks and the temperature of standing water tends to rise with depth until it reaches that critical temperature. (Meanwhile, turbulence is friction and turns the energy of the moving/falling water in a stream into heat, making up for losses. Tap the running water and you get something where you have to pull out a bunch of heat before it even begins to crystallize, followed by a LOT MORE to get it to slush and ice up. If your run isn't enormously long and surrounded by something like moving cold air it won't lose enough heat to stiffen up on you. So just insulate it and keep it moving.