How change circuit to get 15VDC@1A rather than 15VDC@.6A?
The question is, what don't you need to change.

1) Running the transformer at maximum current (1 A average) is doable, but not a great idea. So go to a 2 A unit. And the transformer voltage is way too low. If you have a scope, look at your output with a 0.6 A load, and I'll bet you see dips in the output which occur at a 120 Hz rate. 18 volts is a much better choice. 

2) 1N4007s are, in fact, rated for 1 A. That doesn't mean it's a good idea. Get beefier diodes.

3) Your 1000 uF capacitor is marginal at 0.6 amps, and probably inadequate at 1 amp. Go with at least 2200 uF, and 4700 uF is even better. Plus, if you boost your transformer voltage, at no load you'll be right near 25 volts. Better go to a 35 volt capacitor at least.

4) With a decent input voltage and 1 amp output, your 7815 is going to dissipate more heat, and without a proper heat sink it will fry. Get a heat sink.

5) Eh, you can keep the 1 uF capacitor.