My PC died a few weeks back, I've replaced it since, but I'm now unsure as to how to push to my repo without having to re-clone the entire repo again on my new PC.
I've created another SSH key and linked to my GitHub.
Since you are talking about pushing, i assume you have the local repo on the new computer and you've installed the git client on the new computer.
You can do a git pull or git fetch request so that you can reconnect to the repo.
After entering the git pull or git fetch command, your computer will ask for your git credentials, after entering them correctly, you will be able to push to your repo