Can I send data to my computer from the serial port on an Arduino/ATTiny?
Neither the ATTiny45 (parameters overview) nor the ATTiny85 (parameters overview) has a UART, which makes it a bit painful to connect them directly to a computer.

If they had a UART a common procedure would be to use a USB to serial adapter, preferably with TTL voltage levels (5V or 3.3V) and use the UART in the chip directly.

In your case it's not as easy as you don't have a UART. They do have both SPI and I2C though so you can make them talk through the Arduino as it has both SPI/I2C and UART support. Another alternative is to use an "USB to SPI/I2C" dongle, but they're not really that easy to use (and not what you're after).

Edit: Another solution proposed by @AndrejaKo is to implement the UART in software. If you're only sending this can be quite easy as you only need to get the timing right and can just bang out the bits from there.