The techincal part is pretty straight forward,. However, prior to starting the upgrade you need to check whether all involved components (e.g. hardware/firmware, plug-ins, add-ons, 3-rd party backup applications, ...) are supported with the new version (see VMware Compatibility Guide). Depending on the installed VMware applications/components you may want to take a look for the proper order in which you need to upgrade them, see VMware KB: Update sequence for vSphere 5.5 and its compatible VMware products as well as VMware Product Interoperability Matrix
Licenses are provided for major versions, so if you already run vSphere 5.1 you will not need to get new license keys. That said, with active SnS (Support and Subscription) you can always up- and/or downgrade licenses free of charge if required.
André