There are two main ways to move from TFS 2013 Update 4 to TFS 2015:
Option 1: In-place migration (Recommended)
With an in-place migration, you essentially install TFS 2015 over the top of an existing TFS 2013 Update 4 installation. The benefits of this approach are that you don't need additional hardware, and don't need to change network settings like DNS addresses to change between environments.
Option 2: Migration to a new environment
Migrating to a new environment may be required if you need to move to new upgraded hardware, or if you are not confident with your rollback plan. The key benefit with migrating to a new environment is that you can quickly switch back to your original environment should anything go wrong during the upgrade.
Reducing downtime
With both options, you can also use the TfsPreUpgrade tool to reduce downtime. As per the article, the TfsPreUpgrade tool will:
By using the TfsPreUpgrade tool, you can thereby reduce the downtime for the actual cut-over to the new version by ensuring that your databases are updated in advance.