I'd advise bigger, even total water changes with dechlorinated water on a daily basis as a temporary measure. The problem metabolite is ammonia, resulting from breakdown of protein. It is excreted mainly through axolotl gills and builds up even if debris is cleared out. The poisonous level depends on pH and temperature but without a filter it could rapidly prove fatal.
Filter is really a misnomer, its a sewerage treatment plant! If you want to run the tank without a filter you will need to commit to daily 20% or deeper water changes indefinitely. Read around housing on the associated axolotl site and this article on the caudata culture site:
http://www.caudata.org/cc/articles/cyclingEDK.shtml
Please avoid the gravel, it chokes axolotls so sub-gravel filters are not wise.
It is possible to run a tank without any obvious filters. I run two tanks absolutely stuffed with plants which metabolize all the ammonia instead of a filter, but it is a tricky setup to obtain and I used a conventional filter in both these tanks for about 9 months while I got the plants established.
Why not buy pH and ammonia test kits (and if the budget will stretch nitrite and nitrate ones as well), preferably the liquid based ones, and see what the results are?
