The 23-year-old completed his permanent move from Motherwell for an undisclosed fee on Thursday and made an impressive debut for the Blades in their 3-0 FA Cup win at Oxford.
"It's great to sign a three-and-a-half year deal here," said Murphy. "Hopefully I can progress as much as the club is going to progress. It's exciting times.
"You can see how big a club it is. Obviously the aim is to get promotion this season and if that happens you never know. You've seen teams do it before. It's going to be difficult this season getting promotion, but if we can do that you never know.
"I'm sure Sheffield United are bigger than a lot of clubs already in there.
"It's everyone's ambition here to get back to the Premier League. Nobody here wants to stay in League One, they all want to push on and I'm no different. You want to push as high as possible and I'm sure we can do that."