OK, a modified connect 4 it is.
Do you have 5 buttons? Have you connected them up and confirmed that you can read each one?
Do that as a separate sketch with no LED work in it at all. Just write to the serial monitor when each button is pressed. Up. Down. Left. Right. Place.
Once you have confirmed that our buttons are wired up and you can read them, start working on your game algorithm.
Something to think about. Your current simple sketch to turns on and off a single pixel with the single button. But it does so by looking at the current state of the button. The entire time your button is pressed, the pixel is repeatedly set on, and the entire time the button is pressed, the pixel is repeatedly set off. I think you will need to change your logic to something that reacts 'when a button becomes pressed' instead of reacting the entire time a button is pressed.
But beginning ideas for you:
Make a row and column variable.
Set them both to zero.
When UP becomes pressed, row++
When DOWN becomes pressed, row--
When LEFT becomes pressed, column--
When Right becomes pressed, column++
On all of those, only if that would not put row or column outside of your 0-5 bounds.
Then turn on the pixel defined by row * 6 + column
This should let you draw on your board.