I would not advise building the power supply from scratch. I recommend buying a pre-build power supply.
The only part I suggested you make, is the board with three transistors, switching the LEDs, because it's much easier to do, than hacking a board. The transistors don't have to be match. They just need to have sufficient voltage and current ratings for the LED strip. Heck do whatever you feel more comfortable with. If you prefer to hack, then go ahead.
The voltage rating of the LED strip can be determined by looking at it an counting the number of LEDs vs resistors. The blue LEDs have a voltage drop of about 3V, with the resistor dropping the remaining voltage. If there's one resistor per LED, then the strip is designed for 5V operation. 12V and 24V LED strip has, 3 LEDs and 6 LEDs, per resistor
The current of the LEDs on the strip can be determined using Ohm's law.
I = (VS-VF)/R
Where R is the value of the resistor. This can be obtained by looking at the resistors on the LED strip. It's normally the standard surface mount resistor code, with the last digit representing the number of zeros and the preceding digits the value, so 101 is 100 Ohms.
VS is the supply voltage
VF the LED forward voltage. Normally about 3 for green, blue and white and 2 for red. Of course this is multiplied by the number of LEDs in series, so if your strip has thee LEDs, per resistor, that means there are three LEDs in series.
That will give you the current through each series string of LEDs.
Define reasonable brightness and size?
Note that higher voltage LED strip is generally more efficient. 5V strip is poor as 3V goes to the LEDs and 2V is dropped in the resistor, so 40% is wasted. 12V and 24V LED strop, drop 3V, or 4V in the resistors, which is 25% wastage. 24V is generally a bit better because the current is lower, so there's less voltage drop along the strip.
If you're using 5V LED strip, don't worry about killing your PC. It's virtually impossible.
If you're using 12V, or 24V LED strip, it's unlikely you'll kill your PC. The driver transistors are connected to the Arduino, rather than directly to your PC, so the Arduinio is more likely to blow up, if the driver transistors fail. Using resistors in series with the gates of the driver transistors (refer to my schematic a few posts ago) it will provide some protection for the Arduinio, against the driver transistors faling.