The goal of a fishless cycle is to give the bacteria that eat wish waste (ammonia and nitrites) time to get established before adding in your betta.
So you need to add something to the tank to feed the bacteria and get the colony started.
There are many ways to do this. Some people use straight pure ammonia and dose the tank every day, some will throw in a cooked shrimp and let it decompose in the tank (IMO one of the easiest ways), or you could add a pinch of fish food every day.
Then you will need to start watching your ammonia, nitrite, and nitrate levels. The bacteria that you need convert ammonia into nitrite, and nitrite into nitrate. Nitrates are less toxic to fish than the other two, and also will be used by live plants as food. At first, the ammonia levels will peak, then you'll start getting some nitrite levels, and then you'll get some nitrate levels.
The whole process usually takes at least a few weeks (a month is not unusual), but once the water has stablized at 0ppm ammonia and 0ppm nitrite but there is a nitrate reading, you're ready to add your fish.
NOW- on the other hand- if this is a very small betta tank (say, 3gal or less) you might not need to cycle it at all. Small tanks can be difficult to cycle, and many people make water changes instead of even trying to cycle the tank. Depending on the size of the tank, a small daily or even weekly water change may be enough to keep the ammonia and nitrite levels under control, especially if the tank has live plants.
As lauraleellbp said. If this is a small tank or betta container there is not enough surface for it to really build enough bacterial colonies. In this case it´s better to keep up with the water changes to clean up ammonia/nitrite buildup in your fish´s water.
Bettas are very hardy fish that tolerate amazingly the waste, but you will learn that when your water is polluted he becomes very inactive, and as soon as you change the water he becomes active and swims around. Bi-weekly water changes will keep a fish happy and active
I've got a cycled 2.5 gallon on my desk at work - looking at it right now as I practice my touch-typing, as a matter of fact... (So damned s-l-o-w! This post may take me 20 minutes to finish!)
lauraleellbd covered all the correct points. Set your tank up exactly as you want it (except for the fish, of course), and add some ammonia. Two separate colonies of bacteria will develop in the gravel and filter. The first will eat ammonia and excrete nitrites. The second will eat nitrites and excrete nitrates. Live plants and partial water changes are use to keep the nitrates (which are much less harmful to your betta than ammonia or nitrites) at safe levels.
An accurate test kit is essential. Add enough ammonia to get between 5 and 10 ppm and keep it at that level until it drops almost to zero in a 24-hour period. Then test for nitrites (they'll probably be very high until the second type of bacteria gets established). Keep adding ammonia, too (you don't want the first colony to starve).
Eventually you'll get zero readings for both ammonia and nitrites. Be patient - it may take a while. This means you're cycled and converting 100% of them into nitrates. Then do one large water change (maybe 75%) and add your fish. His waste (and any uneaten food) will supply the ammonia from now on.
Plants aid the process, and should be added right from the beginning. A great way to "jump start" the cycling is to borrow a used filter and some gravel from an already-cycled tank. Both will already contain both kinds of bacteria. All they have to do then is migrate into your gravel and filter. This method reduced the cycling process to a week or two.
It's work and it takes patience, but it's so worth it, and so much better for your betta!
And (I hope this is OK, mods) if you haven't yet, check out www.ultimatebettas.com . It's a great site with some very helpful, knowledgeable people (kinda like here...)