Pick transistor to drive a coil from Attiny85 with 2.5 (+/-500) kHz signal (with screenshots)
I am getting an output with noise and it is definitely not a square
  wave.


This is what I'd expect to see: -

So, start at the beginning with the transistor deactivated. 

You then push current into the base from your MCU and pretty immediately the collector voltage falls to a few tens of milli volts above 0V. Within microseconds, this low voltage starts rising towards a couple of hundred milli volts (as one would expect given that the inductor current is rising from zero amps and the transistor's job is becoming more difficult).

Then you switch off the base and the collector voltage rises to (possibly) a few hundred volts except maybe the transistor breaks-down because the back emf (due to open circuiting the inductor). It might damage the transistor but you might get away with it due to a certain amount of current limiting from R4 and parasitic capacitance. 

This high positive voltage surge might last hundreds of micro seconds then all the inductor's stored energy is gone and the collector will drop (whilst possibly oscillating/ringing a bit) back at Vcc (providing it didn't get damaged too much).

Then the process repeats.