Q: How do I make my own Turing machine?

A: Think one.

A Turing machine is the simplest possible computer, and can mimic any more complex computer. It works by having a tape (like a ticker tape) run across the head of the machine. The head reads the symbols on the tape one at a time, performs an operation on that symbol then gives an output. A more extensive informal definition can be found here.

Since a Turing machine is defined by the process above, if you can think a Turing machine (e.g. if you can visualize the symbols on the tape, think of the function to be applied, then visualize the output from that function) then the neurons in your brain that are performing these functions are an actual Turing machine and not just a representation of one.

(Of course, it seems that it would be difficult to maintain a working machine in your brain for a long time, but that makes it no less an actual Turing machine. A machine made of more standard materials would still have been a Turing machine even if it was destroyed after the first computation.)