Your computer does not use a large percentage of an average electricity bill. Upgrading it won't double your bill.
Newer PCs typically use more power than older ones, so there may be an increase in your bill, but this is not due to the PSU involved. A PSU will only put out as much power as is required.
As for reducing power consumption, take a look at the Power Options in the Control Panel. If your monitor is a CRT, having it turned off when not in use is beneficial, as they use a significant proportion of an average PCs total power consumption. Turning it off or putting it on Stand By when not in use will also help.
If you're trying to cut down every last Watt, removing unneeded devices and turning on Cool and Quiet will help somewhat.