To my surprise, the chip was pretty easy to understand. The two metal tabs connecting the chip to the battery were obviously the + and - of the battery, and the board even indicated which one was positive and negative. The lithium battery also had 3.6v printed on it making this extremely easy for me to just use 3 AAA batteries instead. Normally a AAA battery is 1.5v so i chose rechargeable batteries which put out 1.2v each.
Using a AAA battery pack i had laying around, i simply soldered in the wires to the chip inside the PSP battery. I made sure to thread the wires through the other half of plastic from the battery case so i could close the battery back up. Then cut a little slit in the edge of the final battery cover to allow the wires through. After it was all together, i powered on. In the settings my PSP showed 82% battery and allowed me to update my device. I then stuck three little velcro dots to the back of my PSP and the battery case so i can carry it again.