This article is for cPanel Accounts. Determine where your account is with this guide.
This requires a little MySQL trick. It's one simple edit in the database.
First, make sure that your WordPress installation is complete, and that you know what the name of the database is. If Softaculous installed WordPress for you, it's likely the name of the database is something like:
username_wrdp1
and so forth.
Next, log into your cPanel. Select the function "MySQL Databases". Down at the bottom, there is a link to PHPMyAdmin (which looks like a sailboat.)
Once here, you need to select your database name from the drop-down box on the left. (ie "_wrdp1")
This will display the names of the Tables underneath the dropdown box. Click on the one called "wp_options".
Now there is a list of tabs at the top of the page to the right of the drop-down box. The First one is called "Browse". Click here.
Next, the database entries are listed in columns and rows. You want to select the Pencil Icon (for Edit) on the left for the VERY TOP row (option_name is "site URL")
Once you've clicked the Pencil Icon, you will see your domain name listed something like this:
http://ineedtotest.com
You want to change that to something like this:
http://555.123.556.495/~username/wordpress/
This step is crucial so you don't make a mistake. Please take a moment to verify the URL you are typing in here is correct, or you won't see any change.
Once you've changed the URL, down at the bottom there is a Dropdown Box that says "Save". If it does not say Save, change it to Save, then select the "Go" button. After you hit "Go", you're done. Visit your temporary URL for your blog, and all the images will show up just fine.
Follow-up
Once you've changed your name servers so your domain name resolves to your cPanel account, you don't have to go back into phpMyAdmin to make the change again. Simply log into your Admin Panel, there is a tab for "Options". In there you will see:
WordPress address (URI)
Is set to your temporary url. Change this to your domain name with any folders required, and you're back to normal.