Horn switches switch to ground. Power for your original horn really is supplied from a fused battery source, passes through the horn relay, and when you press the horn button the button completes the circuit to earth.

So, you need to wire your relay coil like this -

12 volts from a fused battery source to:

Your relay coil, to:

The horn switch, which then switches to:

Ground.

Just like how your current horn relay works.