I know this is an old post, so I'm not sure if anyone wants this info. I know the tri-border area pretty well. Let me know if you still need help - I have done the trip more than once!
You can only get into this spectacular, tropical and adventurous town from Colombia by plane. The only place connected to Leticia by a commercial airline is Bogota - its El Dorado International Airport has flights to Leticia on Aerorepublica. From Peru you can also take a rapido (speedboat).