To be sure of what you're doing when you want to upgrade, check if your material, devices are compliant with the ESXi version wanted on the below site :
If it's not supported (in your case you have a warning about your CPU) , it could have a lot of misfunction thereafter...
About licences, you can ask evaluation licence for a while then only buy it to a reseller. I don't think you can cumulate evaluations licences
5.5 is End of General Support for almost a year now.
Please see whether you can still register for the free Hypervisor at https://my.vmware.com/web/vmware/evalcenter?p=free-esxi5?
For the required Realtek drivers, see e.g. https://vibsdepot.v-front.de/wiki/index.php/List_of_currently_available_ESXi_packages. Note that some of the drivers may still work on ESXi 6.0.