Shop around for better prices than those I've linked to. For those who don't want to do so, the items I've linked in each below is what I personally used for this project.

Step 1
This step is to prep the electrical stuff for use. The power supply that is linked above has 3 terminals for the (+) and 3 terminals for the (-). We will be using all three terminals for this project. To better utilize the terminals, it will be necessary to cut the plugs off the end of both pumps and all 3 fans and strip the wires back about 1/4 inch to 3/8". They will still work just fine, if you are not comfortable with doing this, then try to find a plug that you can solder to a board and plug them in. Just ensure that the red wire on each fan and pump goes to the positive terminal (+) and the blacks go to the negative terminal (-) on the power supply.
If you want to make it look neat and tidy, getting crimp on terminal lugs for a 22 gauge or 24 gauge wire will be ideal. (I used larger lugs and thus had to fold the wires to make them work)
Step 2
In this step, we will make the preparations for the liquid parts. We will need 3 tubes for each side of the cooler. In other words, 3 tubes for hot and 3 tubes for cold. Starting at the pump, locate the directional arrows on the pump. One should be pointing into the pump and one out of the pump. For the cold side we are going to go from side pointing out of the pump to one side of the aluminum heat sink. From the other side of the heat sink to one side of the single fan radiator and then back to the pump. The reason for this is we don't want to add heat to the liquid before it has the chance to cool the air going through the radiator. We will do the same thing for the hot side with the exception that the hot side will be using the 2-fan radiator. This will complete the set up of the water system short of adding the water. We will do that later on.
Step 3
This next step will get a little messy but is not a hard one. I used a popsicle stick for this part. We just need to put a dot of the thermal compound on one of the heat sinks. Use the popsicle stick to spread it as thinly as you can over it. This will help with transferring heat from the TEC to the heat sink and thus to the water. Just note, the TEC has a hot and cold side to them. The cold side is the side the letters are on. However, if you get it backwards, don't worry it is a very simple fix. Once you have the compound added, set it down and do the other one as well. When both are done, sandwich the TEC between them and keep them as square as possible. You will want to clamp them so that they don't fall apart just don't put too much clamping force or it will break the ceramics of the TEC. Just snugged up a little is plenty, you should see the thermal compound you just added squish out the sides. You can use isopropyl alcohol to wipe it off with a rag if it bothers you too much.
Step 4
In this step, we will be filling the water system up. This can get a little tricky, but it can get done. If you have a funnel, use it. I didn't have one, so ended spilling a bit of water on my work bench. Hold the pump higher than the rest of the leg you are filling and if you can, turn the radiator upside down for now. Start pouring in the water in the hole at the top of the pump reservoir. Once it is filled, put the cap in and move the radiator around a bit to get the air to come out into the pump reservoir. Once air is out add more and repeat until filled. The other option is to run the pump for a short spell and circulate the water to "burp" the system. I installed an on and off toggle switch just for this. It is not required, so it was not included in the parts list. It does make it easier though.
Be sure to fill the other side as well.
Step 5
Now that we have water in the system, we can now hook up our electrical wiring (if you've not done so already in step 1). On the positive terminals we will attach all the red wires. I suggest twisting the fans together for one terminal. Twisting the reds of the pumps together for the second terminal and leaving the last terminal for the TEC. Do the same thing for the black wires and land them on the negative terminals. For the 120V for this, you will need to make a cable if you don't have one. What I did was take an old computer cable for a computer that is now useless and cutting the part off that goes to the computer side and leaving the pronged end. I then stripped it back and landed the black on the L and the white on the N and of course the green on the ground. If you don't have that luxury, you can get a 3 prong plug at your local hardware store and ask for some 3 conductor cable that you can use with it, or buy a short extension cord that you can cut the end off of. (Perhaps I should make this a project as well later on to keep in a project kit)
Step 6
Now is the moment of truth. You just need to plug in your power supply and get the pumps pumping and fans turning. Check back every now and then and you should see one side getting warmer and the other colder. In the section below for possible improvements you will see a few things that can be done to make this better. But don't stop there. Improve on this however you like. Change it enough to make it your own!



Another thing that will also help to improve it is to reduce the size of the space being cooled. I was running this in my living room. For such a small cooler, there is no way it is pulling enough heat to overcome the heat sources in the room. If it were a small cooler, it may reduce the ambient temperature low enough which will allow the water to get colder. It will require another experiment (results will be coming, parts are on order!!).