The first step is to check whether your hosts/hardware is supported by ESXi 6.7 (see VMware Compatibility Guide). Also check whether any 3rd-party software that you may use (e.g. for backup) supports the new version. If everything is ok, you may proceed with upgrading directly to the new version (see VMware Product Interoperability Matrices).
In case you are currently running vCenter for Windows, you may want to upgrade/migrate to the vCenter Server Appliance, because vCenter Server for Windows will not be available anymore in the future.
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Yes, vCenter Server 6.7 can manage ESXi 6.0 hosts, so you will first need to migrate to vCSA 6.7.
For a detailed description, and prerequisites, and the steps to take, please see the VMware documentation / upgrade guides.
If you hav questions about the steps, please feel free to ask here in the communities.