Hello, and welcome to the forums! Im not particularly savy on the GC specifically, but the definitive gamecube trimming guide would be a good place to start, but unfortunately, most of Ashen's pictures have been taken offline. Fortunately, the photos are backed up on GCforever,
linked here. It has several reference diagrams that contain locations where you can tap into those voltage lines.
However, nobody makes gamecube portables anymore. The wasps (modchips) required to make gamecube portables are becoming incredibly rare, and the wii was found to be a far superior option for portables. It runs all gamecube games flawlessly, uses a fraction of the power, generates a fraction of the heat, don't require modchips or memory cards to be wired up, are easier to wire up, are more rugged, cost less overall, and can be cut much smaller. They also play wii games, virtual console games, wiiware, homebrew, and better emulators. While you are free to make a gamecube portable, it isn't a wise decision given how much easier and better a wii portable is.