Assuming you just have the A1200 motherboard and the BlizzardPPC+RAM, you can ensure that enough power is fed into the system as follows:
Splice an original A1200 power cable to the Tower cases's AT/ATX Power supply. There are plenty of documents online describing how to do this.
In theory, you can connect the spliced 1200 power cable to the original power socket on the back of the A1200 and that is the end of it. However, the BPPC and up to 256MB of RAM can draw quite a lot of current, sometimes too much for the single connection above.
A solution to this is to basically use the A1200 motherboard floppy power connector to provide a second power feed. Instead of having your floppy power cable connected to it, you can connect a floppy power cable from the Tower case's PSU to this port making exactly sure you have connected it the right way round otherwise you will cross the 5V/12V rails and that will not smell nice at all.
Together, the two power feeds (original cable and floppy connector) ensure that enough current gets into the system without all of it having to go through a single route. This is how my 1200T is powered.
You might also consider getting a different cooler for the PPC card. The stock one is not so great.