A SharePoint blog site is just a site built using the blog site template. You can use that as a starting point to build something specific to your requirements. In your case it sounds as though you need to create a custom web part for just the category list (which should be fairly easy if you have some development skills - Categories is just a list in your site), and everything else is fairly standard. Before embarking on your development project, see if you can achieve what you want with a bit of clever XSLT in combination with the existing XsltListView web part.
Whichever way you customize the category list view, you will want to modify the various pages that exist in a blog site. The key ones will be the default.aspx page in the root and the Post.aspx page associated with the Posts list. The category list is in the PlaceHolderLeftNavBar content placeholder. The best way of doing this is using SharePoint Designer (preferably not directly on your production server).
Once you have made your modifications you can save your changes as a site template for re-use or deployment to your production server.