Is it legal to breed dogs in the backyard of a condominium?

Breeding dogs may or may not be a business. The 'backyard' may or may not be common area belonging to all owners, or limited common area owned by all but available to not all owners.

Read your governing documents to determine whether or not you can operate a business from your unit.

Also, read your governing documents to determine whether or not you have complete ownership of the backyard, or whether it also belongs to other owners.

Finally, if you can operate a business and have complete control of the backyard, you may want to alert the board--and your neighbors--as to your operation, so that should someone complain, it is clear that you are operating within the bounds established by the association.