How can I clean frozen (raw) chicken juice out of my freezer?
It seems that your concern is just that the chicken will defrost before you can finish cleaning the freezer. It won't. Just put the chicken and anything else from the freezer that you want to save in the fridge. Wipe the inside of the empty freezer down with warm soapy water. You can also add bleach in the concentration of 1 TBS per gallon (16ml to 4 liters) to water to disinfect, at that concentration you don't need to rinse. The whole process should take less than 10 minutes. Wrap the chicken a bit better this time, and replace the contents of the freezer. Start to finish, it should only take 10 minutes or so. Even if you don't have room in the fridge, your freezer contents won't defrost on the counter if you hurry.

If you need to defrost, that's a different story. Your best bet then would be to borrow a friend's freezer or use a cooler. You can speed up the process of defrosting with rubbing alcohol. Since manual defrosting would require unplugging the unit, you'll want to move as quickly as you can to protect the contents of the refrigerator, if it's a single unit.