I have a site running ghost and an SSL from name cheap installed and working.
The cert is soon to expire and I have paid to get it renewed, but I am a bit lost as to what next.
Do I just delete everything and go through the same set-up as the first time round, or is there an easier route?
You would indeed need to generate a new CSR and go through the process of submitting the CSR to your SSL provider, awaiting approval from the issuer and then copying the newly created files to your server. At the server level, unless you need to merge the provided files, the only thing that should be required is a copy & paste followed by a restart of the web server (Apache, NGINX, etc).
The expiration dates for certificates are hard-coded when the issuer generates your certificate, which is why this process is (currently) required.
Like you said, you'll have to go through the process again. Expiration dates are hard-coded in certificates, so you'll have to generate a new certificate signing request (and optionally a new private key, although you can use your existing one) and send it to NameCheap. They'll generate a new certificate you can install on your server.