1. Buy some neopixels. You can buy strings, loose pixels, a pop-apart board of pixels... there are tons of options. I really like the WS2812B, they are resilient for thermal, handling rough, and in general being a doof when building something with them. Plus, they're 3 wires in, 3 wires out.
2. Pick up a power supply. One single WS2812B draws about 60mA at full bright white. Multiply this by the number of pixels you want to use, and you have your estimated peak current draw. I feel better erring on the side of caution. With 25 pixels drawing 60mA, we're drawing about 1500mA or 1.5A... I am using a 6A 5v supply for my matrices. They run cool for HOURS and don't drop green/blue near the end.
3. Your favorite uC. I like Arduino. I'm a sucker for easy. The Arduino Uno R3 is used in these matrices, technically it's a clone. They're awesome little boards. The truth is, anything with GPIO that can handle the bitbanging for these little strings of lights would work. An rPi would probably be better suited to large displays.
4. Lots of soldering. I ordered little pop-apart boards of neopixels. Laser-cut into circles with nearly 0.10 spacing pads on the back. Add pins, add protoboard, and now I have little "quickchange" modules if something gets broken. They're seriously tiny, so consider something as a substrate to mount the pixels to.
5. Software, software, software. This is part ofwhy I used Arduino. See #3, I'm a sucker for easy. The Adafruit neopixel library is awesome, and based on their strand-test program, I've developed some nifty patterns for this thing.
6. DMX Shield. Since I've already said I like easy, we're going to use the MAX485 (or a compatible transceiver) and build a small protoboard shield. There are pre-built versions available as well.
7. Tinker and tune. I've found that It's a good idea to leave my USB cable attached to the Arduino. Obviously, this will not work if it's also connected to RS485, but whatever. If I'm uploading new code to the fixture, I won't have it hooked into the DMX512 sender. Fine-tune and develop what you wantit to do, upload, and enjoy

One last tip... If you daisy-chain in an S configuration like I have, you may find "drawing" to this thing tricky. Ensuring you've picked the right pixel by its number can cause headaches (or it did for me)... so, I fired up my favorite spreadsheet application and made a little number grid: