Dog poop pick up: use a bag
The best way to remove dog poop from carpet begins with the most obvious step of getting the bulk of the dog poop picked up. If there is a good deal of bulk to the poop, I prefer to use a leftover plastic grocery bag to pick it up, then invert it and tie it off and throw it in the trash can. For the remainder, or for smaller beginning piles, baby wipes or a paper towel doubled over and then pinched to pick up the poop should do just fine. Its better to pick it up as soon as possible to avoid the stains setting in. Once the bulk of the dog poop is picked up, use another fresh paper towel and pinching motion to help get even more poop out of the carpet.
Dog poop cleaner spray - Don't do this
The more interesting part, and probably the reason you're looking up an article like this, is the cleaning solution to use. First lets talk about something pointless that you shouldn't do but is popular on the internet. Many articles will recommend you mix one part dish soap or detergent with one part white vinegar and then add water to make your cleaning solution. This unfortunately is counterproductive. The reason white vinegar can be effective as a cleaner is because it is highly acidic (pH of 2.5). If you mix this with liquid dish soap and its associated chemical stabilizers this acidity can become neutralized. Some articles also recommend mixing equal parts hydrogen peroxide and liquid dish soap. This makes more sense but can also turn into a problem, because depending on the chemical make up of the liquid dish soap, the highly oxidative hydrogen peroxide can form chemical reactions with it to produce toxic gases (possibly explosive) and caustic byproducts, like if you mix it with a bleach containing detergent. Some articles also recommend not rubbing in any sort of solution and instead just blotting and scraping the top. This makes sense for a drier poop that has not been setting a long time, but if you are looking up an article on how to clean dog poop out of carpet, chances are you have a more serious problem and the poop is a little moister and has set into the carpet a bit. In my experience this blotting is a bit unrealistic, as blotting does not get in between the carpet fibers and only brushes the top.
Best dog poop cleaner solution
Instead of this internet pop-science non-sense, I recommend a different approach. First, mix some liquid dish soap or some powdered dish detergent with water (one tablespoon detergent to one cup water), poor it on the remaining poop area, then rub it in with a brush, then let it sit for 10 minutes. The soap will help break down the cell walls and combine fats with water for removal. After it sits for a bit, wipe it up with a paper towel, rubbing it in, then pour some water over it and wipe that off with another fresh paper towel, a sort of rinsing action.
THEN, you can mix one tablespoon hydrogen peroxide with one cup of water. Do a test blot on part of your carpet that you don't care about and let it sit for a minute to make sure it want damage or discolor your particular carpet. After this check, pour it over the stain and let it sit for a few minutes. While it sits the hydrogen peroxide will go to work on oxidizing the shit (pun intended) out of the remaining poop material in the carpet. Notice how we didn't mix the peroxide with the soap. After a few minutes, wipe this up and then do another rinse with water and wipe up again.
Finally, if there is any remaining odor or if you want to just dry the area out, sprinkle a little baking soda over the stained area, let it sit for 10 minutes or more for up to a day, and then vacuum it up or wipe it up.
By separating all these steps and chemicals apart, you are actually being more effective and safer.