Install the mod chip in your Playstation 2. The method of installation may vary slightly depending on the specific mod chip that you purchase, but the mod chip should come with instructions that walk you through the installation process.
Load your burned Playstation 2 game into the PS2 as you would any other game. Your modified Playstation 2 should be able to play the game as though it were an original copy, since the mod chip will override the copy protection that is included with the system.
If you don't want to install a mod chip in your Playstation 2, an alternative exists in the form of Playstation 2 boot discs. These discs run the boot functions that an original PS2 game would have, then let the user switch the boot disc out with the burned game. The boot discs generally require a card that is included with them to be used to manually eject the drive bay so that you can swap out the discs; these discs may not work correctly on the first try and may not function on all Playstation 2 models, however.