Can you? Sure.
You're going to need to put in footings for the new wall (unless you want to rip the back wall off your house, support the roof, and try to lay concrete in between the bracing).
You're going to need to know where all your underground utilities run through. Talk to the appropriate utilities for that.
You're also going to need to talk to the city to see if they have any setback requirements (minimum distance of the structure from the road, etc).
It might be more cost effective to simply turn the back wall into a double-stud wall and insulate it. Or, if you can make it work. convert the outside of the wall into a larsen truss system and insulate that.
If you go double-stud, look at filling the entire wall with dense-pack cellulose. 8-12" of that should kill sound nicely.
If you go with a larsen truss system, open the back wall and reinsulate it with something intentionally engineered for acoustic insulation (Quiet Batt, Quietzone, Roxul Safe 'n' Sound, etc).
You've already looking into sound-deadening doors and windows. Also, look at sound deadening curtains in back.
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