A large glider type of design would be logical. The craft will support it's own weight, the animals only need to pull you forward and fly with you. I'm imagining a gilder where the tow is from animals and not another powered craft.
For most practical limitations you are going to want the largest and fewest trainable animals that you can get away with. A close scenario to this is the dogsled, its easy to see for practicality the less number of animals the better. It would not be easy to wrangle 100 dogs to attach them to a sled. If you have 1000 eagles for example and it takes 15 seconds to put each one in a harness on then from start to finish that will still take over 4 hours for one person JUST to put each bird in a harness. Not even counting getting them in the correct area to attach them to the craft in the first place.
Another huge reason for a larger animal is the larger a wingspan the animal has the more efficient it will be, purely from a physics stand point.Check out this answer Trevor_G. It's more efficient to move a lot of air slowly then a little bit of air really fast for thrust. Thus a bigger wingspan will allow you to fly longer before the animals need rest.
Ideally if you can get three or four big animals you could have them grip the craft directly, but if the required number of animal gets much more then that a harness for each may be required.
I'm not sure if a eagle like animal could be trained to work together like dogs can. From a trainability standpoint this might be hard to do.