I have a '01 Elantra and when the headlights are on and you touch the lever
to turn on the turn signals the dash lights flick off and on. How do you
remove the headlight switch to look at wiring?

You can remove the headlamp switch by removing the steering column covers.

Then simply remove the screws holding the switch in place and pull the
switch out.

I'd suspect your problem is with the headlamp switch itself.