This'll be long, so be warned
Get the video camera. It'll reveal much about what you're doing right and wrong.
Following is how I was taught to breakfall and consequently how I teach it. It seems to work for most people. The trick is to have the presence of mind to not push yourself too far too quickly. My apologies if it doesn't make sense...I'll do my best to make it clear
1) Like Gene said, start on the mat in breakfall landing position. Have a sempai or sensei check your position and correct it if needed. Raise both legs and slapping arm while inhaling. Exhale and "slap" the mat with all three. Your lower leg, sole of foot on upper leg and slapping arm should all hit at the same time. Do this several times then switch sides and repeat.
2) Do the same as previous but alternate side to side with each one. Don't worry about height, just roll to the other side and make sure the exhale and all three limbs are working at the same time.
3) Now start adding some height as you switch sides. You kind of "kip up" with your legs while you're moving to the other side. You should start low and have a sempai or sensei check you landing position and correct it before going higher.
4) Now is when you start to differentiate between types of breakfalls. For forward breakfalls we again start very low. In a kneeling position like for a forward roll. GO VERY SLOWYLY. Put your forward knee on the floor (like moving in suwari waza). Extend your rear leg behind you. GENTLY
place the shoulder of the same side as your forward leg on the floor as you reach under yourself (like you're reaching for the foot of the extended leg behind you). Now perform a roll over the downed shoulder while keeping your back leg straight and land in breakfall position. This exercise is for learning to land in correct position while your feet are going over your head. It can and should be done very slowly with NO
impact to your shoulder. The only "impact" should be when your legs and arm all meet the mat at the same time.
5) Now stand like you're going to do a regular forward roll. The only differences will be that you will let your rear arm point along the length of your rear leg. It will remain there during the roll/breakfall, and you will keep your rear leg straight instead of tucking it. If you keep your leg straight you will end up in breakfall position. The problem most people who can already roll have with this exercise is they try to keep going to get up by raising their torso off the floor. Keep your torso down and have somebody check your landing position.
6) If you have an extra thick soft mat use it for the first few of these next ones until you know you're landing correctly.
You'll also need a partner for this one. Preferably someone who can breakfall so they can check position and whatnot.
Stand at about an 90* angle to each other (your shoulders should make an "L"). Each grab the others right wrist. Your assistant needs to be relaxed enough to allow you to adjust the height of your grasped arm but stable enough to provide good solid support. Move your arm to a height you're comfortable rolling over...think about when you were a kid and would flip over a horizontal bar on the playground, you'll be doing this with your arm acting as the "bar". Go straight over your arm. This is important. If you go off to the side you end up landing weird. Just before you make contact with the mat your partner should give you a gentle tug to help unroll you and flatten you out. They should also be giving a good solid base of support for you to hold on to (if you pull on their arm you should come off the floor instead of them being pulled down). As you become more comfortable with this exercise and you can consistently land correctly on both sides you can begin to raise the height of the flip.
7) Again using the soft mat if available. Perfom a large yokomenuchi like strike that doesn't stop. Almost like you are trying to strike your rear knee with your forward hand. Allow this to carry you over. You'll be doing essentially the same thing you did in step 6 just without a partner. Like Ron said earlier I find that really tucking your head (watching your rear foot helps) and going straight over helps tremendously in landing correctly.
Well, I hope at least some of that made sense
Everything starting at #4 above is for forward flipping type breakfalls (mae ukemi I believe). We use different dirlls for side, back, and forward falling (like a tree)
It's a lot easier to show it to somebody than it is to write about it. If anything wasn't clear let me know and I'll try to clarify. If anything goes against what your sensei is teaching you by all means ignore that bit. If anybody offers something that sounds like it'll work better, use that instead