So the way batteries work is like this.
Let's say that you have 4 AA batteries. Each one is 1.5v with a 2.4 amp capacity.
If you connect them all up, you can do it two different ways. (In series or in parallel).
If you wire the batteries in series (meaning end to end like this)
-+-+-+-+
You add the voltages but the amperage stays the same. So four AAs in series is the same as a 6v 2.4A battery.
If you wire them in parallel
(Meaning side to side, like this)
++++
- - - -
Then you add the amperage but keep the voltage. So in this example, you would end up with the equivalent of a 1.5v 9.6A battery.
The amperage isn't the only thing that determines battery life. Watts is amps times volts, and is what really matters in figuring out what battery you want. Let's say something draws about 10 watts. It would last one hour on EITHER a 1v 10A battery or a 10v 1A battery or a 5v 2A battery.
So you can't run the gamecube on 1.5v, but you can run it on 6v using a custom regulator. So you would need to put them in series to get to 6v. That means it is being powered off of a 6v 2.4A battery. 6x2.4= 14.4 Watts. I would estimate that would power your gamecube for about an hour, if you used a wasp fusion and had a smallish screen.
My particular gamecube uses the disc drive so is a power hog, but it draws 2A at 12v, meaning it is 24 Watts. So your battery pack would power my gcp for just over half an hour.
I recommend a premade battery pack that uses lithium ion 18650 cells, because each one is 3.7v, 3.3A, which is only slightly less than all four of your AAs. I strongly recommend you get a pack that has at least 3 18650 cells unless you really don't care at all about battery life.
As to that website you linked, I didn't mean pay a website to build you one, I meant something like this
https://www.amazon.com/dp/B00ME3ZH7C/ref=cm_sw_r_cp_apa_3t4pybMJ1Q0WJ
If you don't want it to run off of 12v, be prepared for it to draw more amps. The only reason you would run it off of less than 12v is to save space inside your case by having less battery capacity or to let you trim off the on board regulator and replace it.
Also, I don't know much about AA specs, but they need to be able to output a high enough voltage at once to support the gc. If they can only put out 1 A at a time for example, you definitely can't use them.
However, you mentioned that this is a dol101 board, which to my knowledge is really only used because the regulator is already on board, making it simpler. If you don't want to make your own custom regulators (expensive and unnecessary unless you're really pushing modding boundaries), you're going to need a 12v battery.
I just dropped $40 just on regulators and related parts for my wii portable. For gamecube, they really aren't necessary unless you want tiny battery life. (Ex: to get to 12v, you need three 18650s, but if you only have room for two batteries in your case, you could make custom regulators and run it off 7.4v instead. It'll still only last 2/3 as long as the three batteries, but it makes it more convenient in some cases.)
I finished my first gamecube portable a couple months back and am gearing up to do a wii portable soon once all the parts come in. 
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