Does Deku/Midoriya ever use his power without breaking his bones? I’m only on season one, currently, but I really want to know if he does ever use his power without it breaking his bones.
Spoilers Ahead. 

Yes, he was able to. In the later seasons, he kept check of his power and only uses less than a hundred percent to avoid injury. To be more precise, 


  If Izuku does not keep One For All below his current limit of 20%, his body can be immensely damaged by the force of his movements. (Wiki) 


During the Internship Arc, however, 


  he was able to use 100% of his power but with the aid of Eri. As seen in Chapters 157 onwards, this was possible since Eri has the ability to rewind the state of the body of those she touches. Izuku's Quirk destroys his body while Eri rewinds it to a previous state or 'heals' it simultaneously and thus, making him maintain 100% power during this particular fight without breaking his bones or damaging his body.