the original wire for your horn should just connect onto a tab on the back of the button. and if it is like the apc wheel I have the back side of the button also has a metal bar that is bent aroud the perimeter of the button. to check pop out the horn button, use an ohm meter and touch the tab with one probe and another metal part on the button with the other probe. when the button is not depressed you should have no continuity. and when the button is pushed you will see continuity.

once you locat this verify that the oem horn switch for your car is connected straight to ground ( does not go through any other items ) using a wiring diagram, or the easy way since the stock horn probably only had 1 wire to it take the wire and tap it to one of the steering wheel bolts on the main hub ( the 3 bolts that make a triangular patern behind the horn button). you can also use a piece of wire to jumper it to an other ground point in the car.

if you car is on so that the horn would normally work and you hear the horn beep then you have ground side switching. now take that other metal point that you faound on the horn with the ohm meter and connect it to one of the 3 bolts in the hub ( assuming that one of them triggered the horn ).

on my 88 crx
I had the same issue with my wheel as the stock horn was single wire and there was only one tab on the horn button of the apc wheel. heres how mine went. the stock wire connects to the tab, solder a 3 inch piece of wire to the metal part that showed continuity when the button was pushed, the other end of the 3 inch wire got 1/2 of the male female disconnects from auto zone ( make sure it is the kind that already has a plastic or rubber insulating sleeve ), then tak a peice of 6 inch wire and crimp on a ring terminal with a big enough hole to run 1 of the 3 bolts through it, the other end of the 6 inch gets the other half of the male female disconnect, now connect the 2 wires together ( this is the new ground wire) and the oem slips onto the tab, put button back in place, ops check.

if something on your rig is not as described or this doesn't work pm me and I'll do what I can to walk you through.

basically the new wheel is not grounded to the button.