I would imagine that youw ill need approval from:
The HOA - (or your neighbor if there is no HOA)
The City - building permits and zoning
My suggestions would be to first contact the HOA and see what restrictions they have then move onto the city building department.
If you let us know where your duplex is then we could help point you in the right direction. Your zipcode says 22947 and so I'm not sure if you are in Boulder, Co or another part of the country.
In general, expanding homes with an addition in neighborhoods like Country Club in the City & County of Denver is quite easy (given the financial resources to do so). A detached single family home would be easier to work with, but a duplex unit is doable.
There will be most likely be a Party Wall agreement on file with the County Clerk (available through any Title Company) that will detail any possible limitations with an attached unit like this. Party Wall agreements are like little mini HOA"s that govern the use & rights of the property since it is attached (roofs since they are shared in attached units are common issues that arise and need documents like this to govern use, improvements, etc).
Country Club is a great central Denver neighborhood. There are other surrouding and adjacent neighborhoods to Country Club where you can get a Single Family Detached home for less $$'s. Maybe something to consider.
Feel free to contact me if you have other questions.
The City & County of Denver has a Walk-Up Help Desk (located at 201 W. Colfax) that is very helpful and will answer all your questions. They will ask you for your address so they can pull up the property and associated information like zoning for the lot.
Be prepared to have at least a rough sketch of proposed design when you go. Ultimately you will need to file for a building permit and that will provide you with a detailed list of items the city will require to issue the permit for your addition.
Any contractor in Denver will be familiar with this process. I would be glad to refer one to you if you need a reliable and well priced contractor for such work.