Upgrade for VMWare vSphere ESXi from any 5.0 to 5.5 is pretty straight forward. However when going from 5.5 to 6.x, there are some added components like PSC (Platform Services Controller) See here.
There are various methods, however I will outline the steps by mounting a virtual DVD ISO (I used Cisco UCS Server - so my Virtual KVM was CIMC).
To mount the ISO and restarting the server and booting through the DVD, follow my other blog post (You would mount a different DVD image which would have the upgrade vSphere ISO).
Note: Check VMWare Compatibility to check if you can go from a specific source to a destination.
Here are the steps:
Once the DVD starts booting, the following screen would appear:
Press Enter when its fully loaded.
Click the partition where the ESXi resides (normally its the smaller partition - the bigger partition has the virtual machines)
Note: Be careful in this step
Select what you intend on doing. I selected the Upgrade with preserving VMFS (which is the default)
Once complete, the server would go through a power cycle.
Once the server power cycles, it would go to the VMWare yellow screen. Turn on the VMs (using vSphere).
Note: vSphere on your local desktop will have to go through an upgrade once the ESXi is upgraded.