Flying a kite is much easier if you have:
1. A good way to control taking the line in and out -- how about a fishing reel?
2. Leverage to guide the kite and something to absorb shocks -- how about a fishing rod?
I first saw this as a kid at Jones Beach. One day, there were a bunch of people "fishing" with their lines going up into the sky, rather than down into the water. It works great! It's even a nice alternative you can switch to if the wind is good and the fish aren't biting.
Step 1: Tackle Choices
The type of rod and reel doesn't seem to matter much, but here are a few guidelines:
1. Offshore reels work best for large kites, but both baitcasting and spinning reels work well.
2. Conventional monofilament line works fine; avoid tapered "fly fishing" line (no pun intended). Use the lightest line that you're sure can handle the force, usually 12-30 pound test.
3. Especially if you're using a spinning reel, connect the kite with a swivel to avoid line twisting.
4. The rod should be relatively stiff for the best control, but a little flexing acts as a nice shock absorber and makes flying kites in gusty winds easier on you and the kite.
In summary, size your tackle to match the forces that flying your kite will impose. If you're really just taking a kite along in case the fish aren't biting, size the kite to your tackle.
Step 2: Up, Up, and Back
Once your kite is attached, launching it is easy; leave just a little line between the kite and the tip of your rod and use the rod to leverage the kite up against the wind.
Get more line out by either letting the kite pull against the reel's drag setting or reeling the line out. If you let it free-spool out in a strong wind, it can be hard to stop (and an abrupt stop can break your line).
The trick to keeping a kite aloft is keeping a constant pressure against the wind. Gusts can be handled well by setting the reel's drag appropriately, and minor fluctuations in the wind are absorbed by the rod flexing. Dead periods between gusts can be handled by eliminating slack by pointing the rod and reeling in -- actually, you "play" the kite a lot like you would a large fish.
That's also how you bring in the kite. Move the rod tip away from the kite, then reel in as you move the rod tip toward the kite, keeping the force against the kite fairly consistent. Repeat until it's in or the tension on the line becomes low enough to simply reel it in.
Of course, if the wind dies, your kite will simply crash. If that happens, just remember to put some tension on your line as you're reeling in to avoid uneven spooling. I'll usually run the line lightly over my fingers just before it goes into the reel.
That's it. Go fly a kite!