Are you starting fresh, meaning a whole new forum, or just moving what you already have to the new URL?
If it is a new install. remove what you already have. I would remove any add-ons first that you have put in to get it back to base. Then remove the software.
Go to your customer area and set the new URL for the license.
Then install it on the new URL.
If you are moving an existing forum.
change the license
install on the new URL
install any addons
The system should see the database if on the same server. May have to set the config for database communication
May have to copy your old database and put it on the server
Once everything is working delete the XF files from the old URL
At least this is how I remember doing it. Sometimes you can have your host copy all the files over to the new URL. Remember it it always a good idea to back up everything before moving forward.
If in doubt that's a good idea - when you speak to your hosting support first question is do they know xenforo. If so they will tell you whether or not you need to change anything in the config.php file. Even if they don't know xenforo they should be able to advise on that anyway.
I would do this by closing the forum and doing it on a duplicate database. When working open the forum.
The totally utterly crucial thing is to get redirects in place both for SEO (inbound links from external sites and search engines) and your existing users. So redirect the old URL to the new via htaccess.
I recommend setting the new domain in the admin panel before you copy the Xenforo installation. You can do it after, but doing it before will help avoid initial unwanted redirects if you have canonical url enforcement turned on.
Depends on if you are using a panel or not.
If a panel, it's simply create your DB, create the new domain, copy your existing files over to it... import your DB into the one you created for your new domain, make sure your HTTP user has read/write access to all directories in your root, edit your config.php to make sure the credentials are the same.