There have been reports that it's some work to upgrade older versions to the latest and greatest, so you should be prepared for some efforts.
That said, the way you should go is to backup the previous installation (e.g. all directories, database entries etc) and deploy that on your own server. This installation then is updated to the latest version by installing the latest version and pointing it to the data from the previous installation. During the first startup, liferay will (given sufficient privileges on mysql) update the database structure and everything it needs. Keep your backup ready and test thoroughly if everything is upgraded the way you intended it to be.
Also you need to keep an eye on your customized stuff - if you have portlets or other components that use the liferay api, you might need to upgrade those manually to take changed APIs into account.
Theoretically that should be it. I've heard of people having had some problems with this - but it all depends on your level of customization and utilization of features in liferay.
The liferay folks intend to circumvent this in future with their EE environment, where you get better defined upgrade paths and long term support with minor upgrades to your environment, keeping APIs and database requirements stable. I'd hope that even upgrades between major versions will benefit from this, but have not yet tried it.