Looking at the ATtiny core for Arduino 1.6.x it looks like you have to use 'arduino:avrdude' instead of just 'avrdude' in your custom boards.txt files. Sorry I did not know that before.
Without using a custom boards.txt file you can get the same effect (328p with internal clock) by treating your ATmega328p as a LilyPad Arduino board with an ATmega328 processor (Select Tools->Board->LilyPad Arduino, then Tools->Processor-> ATmega328)