2 cups borax
1 cup washing soda
9 ounces laundry soap, like Fels Naptha or Zote (*2)
4 cups Biz powder (*3)
1 cup Oxyclean (optional)
For gel or liquid:
Large pot
1 gallon of water
Additional water (for liquid)
A good soap to use when you live in a household where allergies or asthma are a problem.
I recommend you alternate every few washes with commercial free and clear detergent because it will always do a better job and can help with the longevity of your clothes.
The biggest thing to remember with homemade soap is that your local water makes a huge difference. Try the different types below and adjust ingredients until you find what really works for you.
Powder:
Powder is the simplest method and takes up the least space. The downside is that it clumps easily in very humid air and will need to be broken up again in a food processor.
Grate or cut up the soap. Put some powder, then some soap, then some more powder in the food processor and process until it makes a homogenous powder. Repeat in in batches. Stir together once done.
Store in a bucket or container with a lid in a dry spot.
Dissolve the powder in a cup of hot water before putting directly into the washer if you are using a cold or low warm water setting. If you are using a medium warm to hot water setting, you should be able to just put it in the dispenser.
Use 1/8-1/2 cup depending on your load size and content.
Gel:
This is a little more work than powder and takes up more space. It clumps less easily, but may still do so if it sits for a long time.
Grate or cut up the soap. Put in a pan with the 1 gallon of water and heat on low. Stir frequently.
When the soap is melting, stir in the powdered ingredients a little at a time. Keep stirring frequently.
When the powder and soap are both dissolved into the water, turn off the heat. Allow the mixture to cool completely, stirring periodically.
Pour the cooled gel into a mesh strainer and let the excess water drain away. 
Put the gel through a food processor in small batches to blend smooth. It will puff, so don't try to put it all in at once. Put the resulting marshmallow fluff looking product in a bucket or container with a lid for storage. 
Stir the soap into a cup of hot water before putting directly into the washer.
Use 1/4-1 cup, depending on load size and content.
Liquid:
*Makes more than 2 quarts.
This method requires more work than either of the other two and it takes up a lot more space. It's one advantage is that you can always put it into the detergent dispenser without dissolving it in water first.
Proceed as you would for gel; but instead of draining the water, leave it. As it cools it will thicken. Stir in additional water as needed to keep it the desired consistency.
You will probably end up with double the amount you started with.
Over time, it will separate and you will need to heat it up to re-mix it. You may also be able to mix it with a blender.
Put 1/2-2 cups into the detergent dispenser per a load, depending on load size and content.
Extras:
Put 1/2 cup of vinegar or 1/2 cup of lemon juice in place of fabric softener. (*1)
If you have towels that are particularly stinky, soak them in hot water with 1-2 cups of vinegar for awhile before washing.
For stains, particularly grease, rub Dawn dish soap into the spot before doing a load. You can also wet the fabric and use the Zote or Fels Napta bar on the stain.
Throw 1/2-1 cup of baking soda with your clothes on occasion to brighten them up. You can also put in additional Oxiclean.
If your clothes need a lot of refreshing, dissolve up to 3 cups of baking soda in a sink or bucket of hot water and soak the clothes for awhile before washing.
If your whites look yellowed, use Bluing to correct it. Mix 1/4 tsp. with 1 quart of water. Let the washer fill up with water, then pour it in. Repeat this each time you wash your whites until they look white again.
*Adjust this as needed. In general, if your laundry has:
Soap scum, use less soap and more Biz.
Rust, use more borax in all loads and/or add Iron Out to white loads.
Mineral deposits, use more washing soda. (Keep in mind that Biz contains washing soda, so if you plan to add more of it, you may not also need more washing soda.)
Smells, use more Biz or alternate with commercial detergent more frequently. You can also mix the two for loads.
*1 Manufacturers of washers and softeners will tell you not to use vinegar because it will dissolve your seals. This is only a problem if you have a very old washer or you plan to use an entire bottle each wash. They of course want you to buy softener.
*2 If you have bars of unscented or lightly scented body soap you want to use up, you can replace a small amount of the laundry soap with those.
*3 If you cannot find powder Biz, only liquid, omit it when making your laundry soap. Add the amount of liquid recommended on the bottle of liquid to each load individually.