Running Arduino on an ATTiny45 of 85 chip is a huge cost-saver for mini-projects. However, you cannot connect a HD44780 compatible LCD display as you require 6 outputs, but you only have 5. This issue can be solved by using a shift register to control it using only 3 pins! That means you still have 2 more left over for other use!
Step 1: Parts needed
Here is a list of the parts you will need for this project:
1x Arduino Microcontroller (to act as the programmer, you could use an ISP to program them though.)
1x ATTiny45 / ATTiny85 microcontroller
1x 74HC595 Shift Register
1x HD44780 compatible LCD
1x LED (for testing the microcontroller)
1x Resistor for the LED (use one that is right for the specs of your LED)
1x Potentiometer for LCD contrast (You can connect the contrast to ground
Loads of wires!