Starting hot pepper seeds in a wet paper towel enclosed in a ziploc bag has been the most reliable method for me over the years. I have been using this method since I first started growing hot chili peppers in 2002. I find it provides the best germination rates and quickest germination times.
In the image above I have placed 4 seeds in the center of a piece of paper towel. Some people prefer coffee filters as once the seed sprouts it can start to root into the towel. I just tear or cut the towel away from the root carefully. The coffee filter just didn't seem to absorb water as well and didn't like it myself. Some people recommend soaking seeds in 3% hydrogen peroxide for about 10 minutes first to eliminate any fungus or diseases that might be residing on the seeds. I have done this a few times and its probably not a bad idea, but most of the time I have just put directly in the towel. This may come back to haunt me some day, but its really your choice.
I wet the towel pretty good but not so its dripping wet. I think there is a lot of leeway here, just don't want a lot of pooling water in the bag, but most important that the bag does not dry out ever. I usually cut one paper towel into 4 or 5 long narrow pieces and fold 4 times with the seeds in the middle. Then I place in a ziploc back like shown above. I normally used the snack size ziplocs to save space and have less air inside. Also as you will see in a later picture I fold them in half and put in a tray so the seed area of the fold is always down near the heat. I will reuse the bags sometimes and put 2 varieties in each bag.
Now I place them in a seed starting try. Can probably put them in anything you want, this is something I just have handy. This is a 10" x 10" tray I am using.
You have to get a heat source to the seeds. I use a standard seed starting heat mat. You may read a lot of opinions about the heat this needs to be. Generally though most people will say between 80 and 100 degrees. I usually target 85 to 95 degrees. If you get to 100 you are getting close to cooking them. I use a cheap thermometer and place it under the bags in the tray and check it after a few hours to see what its reading. If its getting close to 95 or above, I elevate the tray above the mat, which I always need to do. See the picture below for an idea. This may vary depending on how you do this, all thats really important is getting the seeds into the proper heat range.
Here's is some pictures of what the seeds will look like when sprouting and getting ready to hit the dir. This first one, they are barely sprouted, but since I have 4 seeds, I will put 2 each in a cell and surely get one out of 2 of them to grow.
The one on the left below is pretty far along. If I catch them like this usually don't wait any longer.
This one is really just perfect the way I like it. It will most like break through the surface in just a day or 2 now. Was lucky also that it did not grow into the towel. Many times by this point they root into the paper.
As you can see, this one has completely sprouted and separated from the seed. I will put this one carefully into the soil and leave just a small portion above the surface.