Get a cocktail shrimp. Toss it into some pantyhose and let it sit in the tank. That will start the process. Grab yourself an API test kit so you can watch your ammonia, nitrite, and nitrate levels. As the shrimp decomps, the bacteria will begin to populate your tank. You'll see the ammonia go up, then down as it turns into nitrite, which will then go down and become nitrates. Then you can use water changes to bring the nitrates down. You'll want the nitrates to stay under 40 to avoid beginning to be toxic.
In terms of the filter media you are using, there isn't any need to run anything while cycling your tank. You can even turn the lights off until it is done to prevent algae growth. This will save you some money as well. If you can, use a siphon hose to suck out what remaining Purigen is laying around in the tank. As others have mentioned some ideas to help contain the product, they are possibilities. I use a product called The Bag, which is made by Seachem to contain their products. It will not get out of this bag. Just make sure to change out or re-energize your Purigen monthly to keep your water pristine and crystal.