How much current can I supply to power PIC12F675?
You don't GIVE current to the device. The device TAKES the current depending on the equivalent impedance of the device, but in cases such as this, where you have a consumer as complicated as a microcontroller, the proper way of doing things is to read microcontroller's datasheet and see how much current it needs in various states and then make sure that your power supply is capable of providing AT LEAST that much current, but preferably more.

Also I doubt your setup will work. You'll be safer if you have a power supply that can provide higher voltage than 5.9 V. Usually it's recommended to have at least two more volts in front of the regulator. Next, you will need to take into account the regulator as well. It has it's own voltage drop which you should find in the regulator's datasheet and there you'll also find the maximum current it can supply and how much current it consumes for its own operation. Make sure that you actually read that! If you do have a regulator that can give regulated 5 V from 5.9 V input, it could happen that it can provide only very little current and that the regulator could get fried. Also don't forget that regulators need additional components to work properly and again, you'll find those in the datasheet. 

If you don't want to read the datasheets (but you really really and again really should), you can watch this short video which explains how linear regulators work.