As the CentOS wiki makes clear, there is no supported way to upgrade from CentOS 5 (C5) to CentOS 6 (C6), because there is no supported way to upgrade from RHEL5 to RHEL6. The official rubric is "clean reinstall". Yes, I find that extremely annoying, too.
However, there is a boot-time kernel flag to the C6 DVD (
upgradeany) which will try to upgrade whatever OS it finds on the host to CentOS6. More details about it appear in the above link; some hand-hacking to the
centos-release RPM is required before its use, and some care and feeding are needed immediately afterwards.
I have used this exactly once, on a clean and fully-updated C5 install, and it did upgrade it to C6 without loss of functionality. I am planning to virtualise a copy of my main C5 server, and attempt the same upgrade, to see how
upgradeany works on a real-life server, but at the time of writing I have not yet done this.
Richard, since I note that you're a new user I hope you will forgive me if I point out local etiquette about answers to questions: when you're satisfied with an answer, you "accept" it by clicking on the tick outline you see by the answer. This drives the reputation system for both you and me, plus it stops the question hanging around forever in the "unanswered" pool. My apologies to you if you already know this.
Edit: there is a C6 "minimal" CD, designed to give you a basic working platform, which can be found at e.g. this mirror site for the i686 version, and elsewhere on your favourite mirror site for the x86_64 version.
The most important requirements change for C6 is that the CPU must support the pae and cx8 flags, or the kernel will simply not boot; you can check this with
grep flags /proc/cpuinfo on your current server. Other hardware requirements, and a link to the Hardware Compatibility List, can be found in the RHEL6 installation guide.