First things first... You need to make sure that your system is capable of running Mountain Lion. First and foremost, since you're running 10.5, we need to make sure you have an Intel CPU in your Mac. Then it needs to be at least a Core 2 Duo, but more than that you'll want to check Apple's website for a list of older models that were left behind at 10.7.
Beyond that, there is no real upgrade path. You can try upgrading, but there's a very good chance that it will not go well, so be making with the backups and make sure you are ready to do a clean install. Also, make sure you don't have any old PPC apps around that you cannot afford to live without. Starting with 10.7, Apple dropped all support for Rosetta, which provided PPC binary compatibility. So far as I know, no one yet has developed a third party solution to that.
After you've done all of the above, you can start worrying about getting a copy of Mountain Lion to install.
Need step by step guide to upgrade from os x 10.5.8 to new os mountain lion

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