Based on your description of your home, the LEED-H reference guide would describe your duplex as a single-family home as it shares a common vertical wall with another home and does not share a common floor or ceiling. As a result, your home meets the requirement for the first of several criteria.
Since your property is a preexisting home, you must plan a major "gut" rehab of the space. From the LEED-H reference guide: "To be considered a major rehab project, the home must be stripped to the studs on at least one side of all external walls and the exterior ceiling, to expose the interstitial space for insulation installation and inspection." This requirement means that you would need to remove all the drywall and insulation on your exterior walls (the common wall can remain intact). It also means that unless your attic space above your home is a sealed attic assembly or you plan on insulating the roof rafters rather than the ceiling lid, you will have to remove that as well.
Next, and perhaps the largest requirement is that your home meet all of the prerequisites of the LEED for Homes program. Download the checklist and investigate the prerequisites for all seven categories to see if you already meet the requirements or can make changes to satisfy these prerequisites.
Lastly, one of the first action items once you have established that your project meets the scope and prerequisites of LEED-H is to contact a LEED for Homes provider. They can assist in answering questions you may have about the checklist and how your project fits into the LEED-H program. This is also the first step in moving forward with any LEED-H project. In addition to helping you analyze your project, these experts can provide guidance on local and federal incentives to help fund your project. The small investment in hiring a provider for an initial meeting could be paid for several times over with these incentives even if you decide not to pursue LEED certification.