how do you win with only a king left?

You cant.

this is all the makings of a great argument.

and you can totally win doing THAT. 

be loud and proud!

Get the king to the other end of the board and make it a queen. Duh.

Ask nicely for your opponent to resign.

In draughts (checkers) you can win with the 'move' (Like zugzwang in chess !)  or of course if you have captured all the opponents pieces.

Interestingly in draughts you win when all the opponents pieces are gridlocked. So to use chess notation if you have a white king on c1 and the opponent has a king on a1 with men on b2 and a3, with him to move he has no moves and so is lost.

Is it possible to win on a rules technicality? Say for example, if your opponent breaks a fide tournament rule, during the last several moves.

If you are rated 1300 you can beat a 2700 with just a king. 

What a patzer comment.

Your opponent needs to have a rook pawn, a horrible positioned king, bad luck,  and be an atheist so you can mate him with 1 knight.

When Im up a knight I usually like to slowly grab a pawn from the table and gently put it back on the board, most of the time I fart so my opponent gets distracted and does not see it happening.

Here is one of my games, I was lucky I ate beans before the round because this fart was explosive and allowed me to place a pawn back on the board.

Yes you can if your opponent accidentally tipped over his king.

Then you're an accidental chess master, ACM.

Or another way is, tease your opponent if he could already say " I resigned". then tell him that the moderator and you will be witnesses that he could really say it smartly.

Then you would be a swindler chess master, SCM.

*Then you would be Frank Marshall*

stalemate

you can do that?

you hope your opponent plays 2 illegal moves, like teleporting his king to the other side or something