Is it possible to program an Attiny85 with a USB-to-TTL converter?
Yes, it is possible.

Note that the FTDI chip (assuming that is what you have) can be accessed as either Virtual Com Port VCP or D2XX. Where VCP is the traditional method of using them as Serial Ports with avrdude calling "-c arduino -P COM3 -b115200". However the later D2XX method allows for the pins to be Bit Bang'ed.

One can see in this tutorial it is possible to call avrdude with a programmer of "-c ftdi". Assuming that the D2XX drivers are installed.

Where I see in the current releases 1.0.5r2 and 1.5.7's avrdude's config files that a programmer of type "avrftdi" is supported, as opposed to the article "ftdi"

However, that is not to say the IDE natively supports its. You will have to either call avdude manually. Or add the "avrftdi" to "\arduino-1.5.7\hardware\arduino\avr\programmers.txt" so that the IDE will be aware of it.



That all said I see "Tiny AVR Programmer" that are very in-expensive.