I laid out my pattern with a measuring tape and sharpie on the the brick mortar ( I found it was easiest to drill into the cross sections of the joints, but not required, so I tried to measure it where the lines interesected at a joint).
I drilled holes with a 1/4 inch drill bit into the intersections. I used an electric drill, not a battery pack, so that I could get enough torque to drill into the mortar -- it took some effort and had to have my teenage sons put some muscle behind it for some of them.
Next, I purchased #10-14 x 1.5 inch Lead Anchors from my home improvement store (they come as a set of 10 anchors and 10 screws, I'll save the screws for a different project). I gently tapped the anchors into the drilled out holes--be careful, because they are soft and bend easily-- it was like using a drywall mount to hang a shelf. Some of the anchors would only tap in about 3/4 of the way and stick out beyond the brick surface, but that was ok, when I screwed the eyebolts in, it pushed the anchor further into the hole.
Screw the eyebolts in next. I had to use a screwdriver through the "eye" to get them tightened all the way in.
I planted 4 Confederate Jasmine plants for $5.99 each during the rainy spring we had this past April/May and they have taken off like wildfire! About every other week, I wind the new growth around the appropriate wire and it is filling in nicely. My goal is to have something like the picture at the top on my brick wall.