Update first to 5.4.2.2 before proceeding to any of the newer versions.

But here's how to do it (Googled "concrete5 manual upgrade"):
http://adamjohnsondesign.com/blog/upgrade-concrete5-with-minimal-do...

And just to highlight few of the most important things in this:
- Always, REALLY ALWAYS do a backup of the current state, the whole public_html folder (if that is your root folder) + dump your database
- Again, I repeat the previous, people tend not to listen to that tip and then they are screwed after swamping the core and running the upgrade scripts
- Again, read the first one, with that you can always revert to a working version if something goes wrong
- Just a glance on the first one again
- And also remember to run the /tools/required/upgrade after you've replaced your core version with the new one

Mainio.

Thank you, the links explained everything very well. It was a lot simpler than I thought.