First, set up the generic nameservers in Namecheap to point to your hosts nameservers that you have identified.
Then go into your hosting control panel at your hosting provider, which I assume will be at: subdomain.website.com/~username/cpanel
If your host is using CPanel you will have an option to 'Add-On' a domain.
If your host is not using Cpanel, look through the other options on your hosting account or contact their support to find out where to 'add-on' the domain to your account.