SoftwareSerial with ATTiny84 using ATtiny library

daywalkerdha:
No, I'm just trying to use the Arduino board as a serial interface because I don't have a separate one.

That is what I was trying to ask. Got it.

The 84's TX pin goes to the RX pin on the Arduino Uno R3 board which makes the TX led blink on the Arduino board.

I think that's correct.

I also connected a 10µF cap between RESET and GND on the Arduino

The Arduino processor has to be removed (not recommended) or forced into reset (recommended). Remove the 10µF capacitor. Connect a jumper between RESET and GND. (Actually, you can probably leave the capacitor in place.)

As you've discovered, the Arduino processor interferes with serial communications. In addition, unless the Arduino processor is essentially removed from the circuit, it is possible to damage one or both processors (some current limiting resistors will protect both sides).

If you are interested, I have a version of the ArduinoISP sketch that makes what you are trying to do much simpler.