How to fix a pedal bin into a kitchen cupboard as a swing-out bin?
Okay, bear with me here. This may look a bit daunting but as far as the construction goes it's all pretty basic. The precise measurements and movements of things are another story. The depth of the cabinet relative to the width of the can will be important because there must be enough throw between open and closed to make it all work. And the distance the arm can descend has to be long enough to open the lid satisfactorily. The weight of the arm has to be heavy enough in the open position to counter act the mechanism of the can but not be so heavy that it snaps it to and fro harshly. And all this is assuming you have a can that's built like this one and the cabinet it goes in is adequately sized to accommodate all of the parts required to pull this off.

If you have a different kind of can I think the basic concept can be adapted to work for most types. I chose this one because it's actually the trickiest to actuate. If you can't shoe horn a carriage into the existing cabinet around the trash can...this idea might be dead in the water. The depth in particular is critical as it must be great enough to accept the can plus the length of travel the lid requires.


First construct a carriage for your can that rides in and out on some 100lb, full extension, side mount drawer slides like these. 
Remove the hinges from your the door of your cabinet and attach it to the front of the carriage so it comes straight out and back instead of swinging aside.

Next attach a bushing and keeper to the cabinet wall. A piece of 1 1/2 pvc would make a good bushing. (seen in blue) Make sure there's enough space between the bushing and the keeper block to allow the arm to pivot.

Now attach a strap of ballistic fabric (cut up an old ratchet strap maybe) to the lid of your can with a rare earth magnet or some other removable means so you can get the can out easily for cleaning. Run the strap to a counter weighted arm. You might have to experiment with the weight so you'll want to be able to add and subtract mass. Lead weights like these would be great.
 
Attach the arm to the back of the carriage with some sort of articulated knuckle (two eyelets hooked together would work). 
If all goes well, when you pull the door/carriage out the weight of the arm will pull down on the strap, opening the lid. As you close it, the arm will be pulled back in and the lid will close itself as its designed to do.


This all may take a bit of trial and error but I believe it's doable for minimal cost out of materials that could be bought at any hardware store and it requires no special tools. If you do get it all to work please post a follow up, I'd love to see pictures of it in action! Cheers.