In order to migrate your website or domain across the server (From shared server to VPS), Its better to take a full account cPanel backup from your old shared server with the help of cPanel. Either you can send that straight away using scp to other server or manually.
The manual way is, First create backup on the document root and wget to the new VPS.
Move this to VPS home folder (/home)
After this you can restore the package with WHM or cPanel script.
Home »Transfers >> Copy Multiple Accounts/Packages From Another Server
/scripts/restorepkg backupfile.tar.gz (make sure backupfile.tar.gz in /home)
If you haven't installed cPanel in your VPS yet, Please follow the steps to install
wget -N http://httpupdate.cpanel.net/latest
chmod +x sh latest
If you have any doubts in this, Better to check this link cPanel_installation
Now other issue is your DNS. What ever the name server you are using, Point that readily to the VPS(from corresponding registrar end) so that you can change the DNS even after the shared account killed.
Make sure that all the accounts(domains) are working perfectly in the VPS before switching the name server IP. You can check the domains with temp URL to make sure that its working perfectly. Otherwise you may get downtime to your domains.
VPS_IP/~account_name/ from local browser
This is the temporary URL format.
let me know if you need any more advices.