Plug your male JST connector into the female JST on the IN end of your NeoPixel strip. This will help make sure you get the wires lined up and soldered to your Circuit Playground the correct way. Solder the wire connected to the black wire (G) coming from the NeoPixels to the G pin on your Circuit Playground. Solder the other wire (connected to the white wire on the NeoPixels) to pin A1.
Next we'll add the JST connector. It works best to make a 3-way splice. Twist the red wire from your JST connector together with the red wire coming from the NeoPixels, and solder on a third red wire which will connect to the screw terminal.
Do the same thing with the black wires.
Plug your Circuit Playground in with both connectors. If you've already uploaded your code, the lights should come on when you connect your power supply.
If you're reasonably certain you've got everything set up right, but your lights aren't working as expected, the most likely culprit is the screw terminal. Try wiggling the wires connected here, or pulling them out and putting them back in again. These things can be a bit twitchy. I went through three different ones before I found one that worked reliably (so, maybe order a few when you're getting started!).
If that's not the problem, check to be double-sure you're soldering the data wire to the IN end of the Neopixels. It won't work the other way. And make sure your connector is soldered up correctly and that you haven't mixed up the G and Data wires.
I didn't want lights across the end of my table, so I split my light strand in two. If you want to do this, carefully cut through the solder pads between any two LEDs and add a wire extension. Be sure you're soldering from the OUT end of one strip to the IN end of the other -- so, the easiest way to think of this is to keep the wires connecting the two LEDs you just cut between. Solder a red wire to +, a black wire to G and a white wire from OUT to IN.