How can I modify my NIkon D5500 so I can make a vintage lens auto focus?

You can't.

I learned that using a TC-16A AF Teleconverter 1.6X might help ...

Whatever you learned is probably wrong. From my googling it looks as if the TC-16A was designed to turn AIS lenses into AF lenses (i.e., an interim step to add electronic communication and the physical linkage to the AF system in the camera body), and that AF lenses are incompatible with it. And you'll still need a focus motor, which your D5500 does not have. In addition, to work on the modern Nikon dSLRs, the TC-16A must be modified.

Is there another teleconverter that will make old lenses auto focus using my D5500?

No. You're missing a focus motor in your camera body, and in the AF lens. The TC does not have a motor of its own, and expects your camera body to have one.

I'm planning to buy some old AI lenses from Japan because those are awesome.

They are. But they are manual focus. Not even AIS, so the TC-16A won't help you with them. Also while you could get pre-AI lenses and use them, be aware they're liable to damage a higher-end body than a D5x00 without modification (grinding down an arm).

Is modifying my camera a good idea? How do I proceed?

If you don't already know how to proceed, and you don't already have a really good working knowledge of soldering, microelectronics, prototyping, a thirst to figure out how to stuff works and to build things (along with a buttload of money), I'd say it's probably not "a good idea." Because you have to add/integrate a focus motor into the body. And that won't be trivial or easy and probably can damage the camera if you've never done anything like it before.

The least painful and expensive route is more likely to be returning your D5500, and saving up for a D7x00 or used D90 body, and then looking into an modified TC-16A if you get AIS lenses.