If you are setting up a new aquarium with a new filter, using your current tank water really isn't necessary since it doesnt contain nitrifying bacteria anyway. Cycling a tank requires the addition of ammonia to get your bacteria to grow.
I will copy and paste my "fishless cycling" instructions here from my website:
First, set up your aquarium using de-chlorinated water and let it run for 24 to 48 hours. This will allow you time to make sure all of your equipment is running properly. Test the water for all of the above listed parameters. These will be your starting base-line measurements. After the aquarium has been running for the recommended time period, add a small amount of your ammonia source. After 24 hours, test the water and record new results. Test the water every day or two and monitor the changes. When the ammonia level begins to drop and the nitrite levels begin to rise, add another small amount of ammonia to keep feeding the bacteria. Keep testing the water quality, and when the nitrates begin to rise, dose the tank with the ammonia source again. This should be the last dose required to finish the cycling process.
At this point, you should see a small rise in the ammonia level, a small rise in the nitrite level and a continuation in the rise of the nitrate level. When the ammonia reading drops to zero, the nitrite drops to zero and the nitrate is at a measurable level, the aquarium is considered to be cycled. Before you add any animals to the aquarium, do at least a 20% water change to lower the level of the nitrates. Different animals tolerate nitrates at different levels, but a safe rule of thumb is to keep your nitrate level below 40ppm. Typically, a 20 to 30% water change each week is enough to keep levels in a safe range.
It is important to note that while the tank is now cycled, the bacterial population will need to grow and compensate for your axolotl being added to the aquarium. The water should be tested on at least once per week to monitor water quality and hopefully prevent potential ammonia and nitrite poisoning.