Help freezing chicken breasts
From your description, these are IQF (individually quick frozen) chicken parts.  They are blast chilled at the processing plant, before being put together in the bag.  They don't stick together because they are so cold that there is no liquid water make them stick.

You can repackage them into any freezer safe set of packages that you prefer, in portion sizes that you prefer.

I suggest using small zip-type freezer bags, with one meal's worth per bag.  You definitely want the type that are rated as freezer bags, as they are less permeable to gas and water vapor, and will keep the the contents in the best condition.  You can squeeze out the excess air, but I think using extra foil or trying to suck out air with a straw is more than you need to do.

(If you really want to ensure there is minimal air, place each bag in water after you fill it, letting the water squeeze out the air, so you can zip it up.)

If you use place all of the small bags in a much larger freezer bag or a sealable freezer safe plastic container, you should get very good protection for each piece, while still being able to use them in small portions.  

The extra layer is just for additional protection, it is not strictly necessary.