There are computers with Atom processors that can be powered off of about 50w but they aren't going to be gaming beasts.
I am going to assume that you want a lot more power than something like that will give you.
I will try to make a light gaming build that still goes pretty light on the power bills.
If all you want is to get no the internet look up atom processor computers, otherwise, strongly consider the below system. You can go on Newegg and just copy and paste these part descriptions into the search bar and it will pull them up.
This processor only uses 65w of power and it is very strong in relation to that power usage.
GIGABYTE GA-Z68AP-D3(R2.0) LGA 1155 Intel Z68 HDMI SATA 6Gb/s USB 3.0 ATX
Basic motherboard, not particularly much different in terms of power compared to other models. Motherboards aren't the best place to think about energy efficiency.
This RAM is from the highest quality RAM maker and only uses 1.35v. You could go with CT51264BD1339 instead if you only need 1x 4GBs (the suggested part number is 2x 4GBs), but I would avoid it. 8 GBs of RAM is the sweet spot for today's computers for functionality. The 1.35v power usage is as low as it goes in a general sense as well.
ATI HD 7750
You can just pick the cheapest option that comes up when you pick this as many different companies make this part and the more expensive ones are usually ones where a company has tried to squeeze a tad bit more performance out of the card at the cost of higher power requirements.
The 7750 uses a 28 nm manufacturing process which makes it the lowest power using card out of other cards of similar power level. This card only uses the motherboard slot for power which puts it squarely in the low power camp. It can't pull more than 75w even if it wants to because that is all the motherboard will give it. Cards with PCIE connectors can pull greatly more. Potentially 600w from cards with 2x 4 pin PCIE connectors.
Western Digital Caviar Green WD5000AADS 500GB 32MB Cache SATA 3.0Gb/s 3.5" Hard Drive -Bare Drive
This hard drive spins at 5900 RPMS which saves power over the more common 7200 RPM drives, but you take a performance hit because of it. That hit is slightly decreased by the fact that it has 32 MBs of cache rather than the more common 16 MBs or 8 MBs for drives of this rotational speed.
This DVD drive is just the highest quality drive in its class by far. Not much power difference between DVD drives so not optimizing here.
Antec EarthWatts Platinum Series EA-450 450W ATX12V / EPS12V SLI Ready CrossFire Ready 80 PLUS PLATINUM Certified Power Supply
PSUs are one of the top 2 items that you don't want to sacrifice on quality on (along with RAM) and this device is of the highest quality. That means it won't break and it won't break your other stuff.
You can get similar quality PSUs that are less efficient than this for like 1/3 of the price, but if you want the smallest power bill possible you have to pay for a low total wattage platinum PSU and this one fits the bill.
You want low wattage rather than high wattage, because PSUs are most efficient at about 50% load and the load this system will put on the PSU is pretty small. To be 2x of that (so load is close to 50% of the PSU's max) you need a really low wattage number, like the above 450w.
Cooler Master Elite 430
This case is a good entry level case. The important thing is to have a bottom mount PSU rack, because that will keep the heat inside the PSU down. Heat inside the PSU makes it less efficient and damages the internals of the PSU over time.
It only comes with 1 fan standard which is good for the power bill, since more fans mean more power.
However, I want to note here that more fans are "worth it" for most people, because they do a good job making the computer run cooler which keeps parts from suffering heat damage.
With this low power system one fan may be good enough, but there are options to add more if you want to.
Personally, I would suggest you take the small power hit to add 2 x 120 MM fans to the top of the case. It will do wonders for the operation of your computer if you do, and won't add too much to the power bill.
Anyway, there you have it. That is how I would build a computer that wants to cut the power bill as much as possible, but still do everything a computer should be able to do including run games on reasonable settings.