I would like to add overhead light in my kitchen by tapping onto an existing light. The problem is that the circuit is pretty loaded as it is now. If I switch all my existing lights on that circuit to LED do you think I could run another wire?

Have you had problems before with that particular circuit? If not changing one light fixture for another should cause no new problem.

Have you consulted with an electrician, or is this a do it yourself project?

The math should be easy here...
What size are the lights you could replace with LED? 60 watt? More?
Most LEDs are about 13 to 18 watts so each change out may gain you around 40 watts of extra capacity...
What do you think? Do you have a count of fixtures to replace?