I always build on a breadboard before putting anything on a PCB. I recommend you do the same. The red and black wires in the image are simply to show where the relative positive and ground voltages are. This is not a clean signal by any means, but all voltage is now positive, so we do need to keep the two wires straight.

The LED strip has little copper pads spaced between every three LEDs. Cut it just to the left of those pads, cut off the plastic coating and tin the pads to make soldering to the wires easier. Make sure you tin the wires too. In my application I used two short pieces of the LED strip on each side of the cabinets. I had to turn the strip over and very carefully scrape away the plastic coating to expose the copper underneath. I then soldered to the back side of the first length of LEDs and then connected that to the front side of the next length.

You can connect several individual lengths of LEDs to the same output connections on the bridge rectifier. I made four different lengths of 18 LEDs, then put two together end to end to make two lengths of 36. Those two lengths are then connected together at the rectifier output, one strand off to the right, the other off to the left. Just keep in mind the power requirements for each LED and the limitations of the parts, specifically the diodes.

The last image shows the connections all nice and finished with the heat shrink. Be careful when handling the LED strip with wires soldered on. I had one of the copper pads break off with the wire and I lost connection. I was able to fix it by attaching the wires to the back instead. So get the heat shrink on as fast as you can to give the joint some strength.