A filter for a 5g won't be enough for a 20g.
Cycling takes up to 2 months to complete.
Fishless: Get a bottle of ACE Pure ammonia and an eye dropper. You also need a reliable drops kit for ammonia, nitrite and nitrate. Add 10 drops pure ammonia, wait 24 hours and test for ammonia. If it is at 4 ppm, wait another 24 hours and test again. If it's less than that add 10 more drops, wait 24 hours test. Basically you want to test every 24 hours and add another dose any time you see it below 4ppm. After about a week you can start testing for nitrites. The nitrites you ignore, but just keep track of. One day, after 24 hours, ALL ammonia will be gone to zero and you will be left with only nitrites. At this point, dose half - 5 drops a day to keep nitrites around 4 ppm. One day all the nitrites will vanish overnight to zero and you will be left with only nitrates. At this point I would keep dosing an extra couple few days to make sure all ammonia and nitrites still disappears within 24 hours - actually at this point it should all be gone by 12. Then you do a HUGE.. like 99.9% water change with your siphon all the way down to the gravel, but don't disturb the gravel. Fill it up with same temp water use the in tank thermometer to match temp, mix up buckets with conditioner than add so you don't shock your bacteria. Test your nitrates. If they are 5ppm or lower you're good. If they are still too high (mine took a 90% and 2 or 3 50%s) do more water changes until they're down low. From here fish can be acclimated and a once weekly 25%-50% with siphon should keep nitrates <20ppm.
Fish-in:You need to be testing daily with a reliable drops kit for ammonia and nitrite and doing an extra 50% change any time you see either. In addition to this a weekly 50% with siphon is needed. First you will see ammonia, then nitrite. Eventually, hopefully, you will see ammonia fall and stay at 0 even after a week of no water changes, and finally nitrite. At this point you will be left with only nitrates after a full week of no changes and these can be kept <20ppm by weekly 25-50% change with baster
You should let your tank sit with filter and heater and everything 24 hours before adding fish, but this won't cycle the tank.