There are guides and videos showing the process of opening many retro systems. Here is a good one for an NTSC Colecovision.
This guide shows the process of installing TMS-RGB in an NTSC (Rev.H2) Colecovision using a direct, 8-pin mini DIN jack intended for SCART output. If you're installing this in a different retro console/computer or are using a different output jack, your steps will be a little different, but not by much. I'll try to mention differences as they come up.
After consulting the connection diagram for your desired cable, measure the distance from the mod's eventual home under the graphics chip in your system to the location you'd like to run the output jack. Cut as many wires as you need to that length, then solder a wire to each required pad.
Often, not all the pads are required.
Using stranded wire will make this much less frustrating than solid core.
Locate pin 1 of your TMS992xA chip on the underside of your system's motherboard, then line it up with the square through-hole in the corner of the TMS-RGB board. Solder each pin.
If you've installed citrus3000psi's ColecoRGB board in the past, TMS-RGB's output pads face the opposite direction!
There is often a white dot on the back of the motherboard showing the pin 1 location. Notice the white dot just off the left edge of the mod board in the video.
Capturing audio out will vary considerably by system.
NTSC Colecovision: use the leg of C88 nearest the RF shield.
If you know the answer for other compatible systems, please let me know and I'll add a line here.
The L9 inductor on the NTSC Colecovision motherboard is a simple, low-quality filter for the original RF output. TMS-RGB contains its own, higher quality, filtering so this component isn't necessary. Worse, the low quality of the original filter causes overshoot side effects that are hidden by low-quality RF video but that can be seen with RGB output.
By replacing L9 with a small bit of wire, these overshoot artifacts can be eliminated.
The traces on old boards like these are very fragile and can be lifted easily. Exercise extreme caution when desoldering and removing components! If this is something you're uncomfortable with, it might be safer to skip this step completely.
Referring again to your connection diagram, solder the output wires from the mod to the jack of your choice.
If your system provides mono audio but the jack has stereo, just solder the same audio wire to both output pins, as shown.
Now is a good time to test things before proceeding. Connect something to your new video output and fire up the system to see if everything looks good.
With tools, wires, and bits of RF shielding around, it's easy to short things out accidentally. Double-check your work area before powering up your system.
This will vary considerably by system and jack.
For mini DIN-style jacks, the easiest answer might be drilling a hole near the back of the case and using any hardware that came with the jack (or hot glue) to secure it in place.
If you prefer not to alter the original case, it is often possible to reuse the existing hole from the RF port by desoldering the old jack and securing the new one in its place. This guide depicts the older ColecoRGB mod but has an extensive section under "Aligning and Mounting the AV Jack" that shows one way this might be accomplished.
Follow the steps of opening your case in reverse to get everything sealed back together.
It's worthwhile to reattach the ground straps.
If you notice any faint noise ("jailbars") in the background of black areas, this can usually be cleaned up on the receiving end.
Framemeister: Adjust the Brightness setting down a single notch.
OSSC 1.6: Menu → Video in proc →
→ R/Pr offset → 110
→ G/Y offset → 110
→ B/Pb offset → 110
Other: Tinkering with your settings a little (try Contrast first) should clean things up.
After all your hard work, now it's time to kick back and play some games on your newly modded system.
Invite some friends! :D