Hi,
As you mentioned, you cannot upgrade ESXi Host from 4.x to 6.x You need to first upgrade 4.x to 5.x and then upgrade to 6.x
Before upgrading to 5 or 6, make sure you ESXi Host is compatible with ESXi 6.
Check VMware Hardware Compatibility list and check if your server is listed there.
Though you will have ESXi 6 License kit, you can download ESXi 5.x and use it for upgrade process.
Only if you need to use 5.x permanently then you need to get your License keys downgraded for 5.x otherwise once you upgrade Host to 5.x, proceed and upgrade it to ESXi 6.x
Backup -
Before you start upgrade process, make sure you have backup of all critical VMs.
Either you can download/export all VMs as OVF or you can use free edition of Veeam to do the backup of VMs
Upgrade -
You can attach ESXi installer ISO to server and boot server form ISO and do upgrade.
During Installation process make sure you select 'PRESERVE VMFS and Upgrade ESXi Host'
if you do not preserve vmfs, all vms will get deleted during installation/upgrade process.
Once you get ESXi Host upgrade to 5.x Upgrade VMFS version so that it can support 6.
Then finally upgrade ESXi Host to 6.x, again make sure you PRESERVE VMFS during installation.
If you have vCenter and Update manager, Use update manage to perform upgrade that's easy way of upgrade.
Thanks,
Haridas