I would really suggest not DIRECTLY hooking up the button to your ignition, you'll blow the **** outta the starter and the button.
Here's some info, first you need a 4 way automotive relay, the relay will have 4 pins (#30, #85, #86, #87).
Off the relay:
#30 pin -> starter wire (sends power to the starter, starts car)
#85 pin -> #4 slot on button
#86 pin -> Ground (good, clean, solid contact RECOMMENDED)
#87 pin -> constant power (even with car off)
Off the button:
#1 slot -> switched power (when key is turned on, this simply turns the light on in the button)
#2 slot -> Ground
#3 slot -> switched power (same as above)
#4 slot -> #85 pin on relay (sends power to the relay, which then sends power to the starter)
Little walkthrough here:
Switched ignition power is sent to Slot #1 and #3, #1 lights up the button, #3 activates the switch (if the key isn't in the ignition, the switch has no power, so it cannot start the car, very important, it must be wired this way and not to constant power as you don't wanna be able to start the car without the key). Slot #2 on the button is the ground for the light only. Slot #4 sends power to the relay pin #85. Now take 12 gauge wire or at least the gauge of the stock constant power wire and attach it to pin #87; take the same gauge wire and splice the starter wire (wire going TO the starter) and attach to pin #30. Pin #86 should be a good ground. All connections to stock wiring should be SPLICED, do not CUT anything.
Basically this is how the switch works:
Key goes in the ignition, turn the key, power is sent to Slot #1 and #3 on the button; slot #1 and #2 turn on the light; slot #3 activates the switch/turns the switch on. Without the key nothing happens.
Push the button, power is sent from slot #4 on the button to pin #85 activating/switching the relay; when this occurs, power is sent from pin #87 to pin #30 which is attached to the starter, cranking the starter; and your on. Pin #86 must have a good ground or you might not send enough juice, it might not turn over.
That's basically it, it can be done, but it's a pain in the ass. If you're not skilled with this kind of stuff I would not recommend taking on this task, you need some basic knowledge/experience AT LEAST. If you wanna do it, make SURE you disconnect the negative battery terminal first.
Good luck, maybe if someone wants to get all the wire color codes I can write a more indepth/specific writeup.
NOTE: I don't take any responsibility if you blow yourself or your car to pieces, this is exactly how I had it hooked up on my Eclipse, my buddies Eclipse and my buddies Integra, all have worked flawlessly for 3+ years to date. It's a basic setup and works the same way on all cars, except maybe those with computer chipped keys and crap. If you've got a remote starter I can't/don't wanna help, too complicated if I can't see it in person.