The standard way to upgrade from one version of Windows to another is to use Change Product Key on the Control Panel System page. There are other ways using the command line, but in most cases they just do the same thing.
But unfortunately you cannot upgrade directly from Windows 10 Home to Enterprise. If you try to enter a new product key you will get an error such as "Error code 0xc004f050".
However, it seems you can upgrade by a two step process. First enter a Windows 10 Pro key. After a warning, the system will proceed to install new components, reboot once or twice and then magic! Now you have a Windows 10 Pro system. It will still need to be activated.
Now you can go back to Change Product Key, enter that Windows 10 Enterprise key and it just works. Since this is a MAK/KMS key it will need to contact a key server.
Assuming you have an MSDN subscription, both keys can be obtained from the MSDN licence key page. If you prefer to use your corporate keys, that should be OK too.
You may also be able to get to Pro using the standard Microsoft key (which cannot be activated), but I haven't tried this. The key is: