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. Many people do not use or like this method, but it is just what works best for me. I have limited space and equipment so I don't have a climate controlled grow tent etc.
The first step I always do is soaking the seeds. I use the little salad dressing cups you can get at many store in the US. I put a number on the lid and then on the ziploc bag I will use put the same number and the name of the variety I am germinating.
Possibly the most critical part of all of this is the quality of the water. I had troubles for a few years after my city switched water supplies. After I finally tested it, I found it to be a PH of 9. I now use distilled water or PH adjusted water in the 6 to 7 range. If you water source is not good I suggest using distilled water as much as possible in the early stages.
The first step I do is to soak for an hour or 2 in hydrogen peroxide. For this I don't dilute, I use the 3% solution that can be found cheaply at most stores. I make sure to stir it up to be sure all the seed surface is covered. I know some people who have even used bleach at this point but I just use the peroxide. This will help control any fungus/mold/bacteria that may be on the seeds. I feel this step to be very important as many seeds at least have some dormant mold spores that may have formed during the drying process which can be days or weeks in some cases.
Next I drain off the peroxide and then soak in a solution of salt petre and water. I use about 1/4 to 1/2 teaspoon in 2 cups of water. Salt petre is potassium nitrate KNO3. I have acquired mine by getting Spectracide brand stump remover which is nearly pure KNO3. Some people don't like or say this isn't necessary. While I agree it isn't necessary I just do it as it seems to help with stubborn seeds and also seems to reduce germination times in some cases. This is just s personal preference as I have used it off and on since 2002 and I feel my germination is better with it. Other people soak in plain water, others use chamomile tea solutions.
I let the seeds soak for at least 20+ hours usually around 24. I also this year have been putting them in the refridgerator for at least half or all of this time. I don't know that it has helped with rates or time, but it definitely has not hurt.
After the soak I then put them in paper towels. I stain out the seeds and rinse them off before placing in the towels.
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.
I wet the towel thoroughly but then compess it after folding to get most of the water out. You don't want it to be too wet, that seems to hinder the germination process. It just needs to be damn but not dry. 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. 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 90 degrees. I usually target 85 degrees or so. 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 90 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 that's 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 dirt. 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.
For some particularly long germinating or stubborn varieties, I have found they tend to germinate better if I remove them from the heat after a few weeks. I have had several in the last few years that were not germinating. Not wanting to give up on them, I have removed them from the heat and just let them sit somewhere at room temperature. They often times will germinate in the next week or so. It has become my standard process to remove them from heat no matter what after a few weeks.
I have also began leaving my heat mat on a timer. I am not sure if this helps any or not, but definitely no harm. It allows them to cool down at night when the heat mat is off more like they would in nature with the day / night cycles.