How do I remove the offset from the DAC output and change the range?
If you want pulses to stay looking like pulses, you need to avoid any circuitry that has any frequency response at all, but this DAC is clearly intended for audio use, where the output would be AC coupled to the next stage by a capacitor, but that's not possible if you want a near-DC response. 
  If the output at a '0' input is predictable at 2V, to get from a 1-3V output to your desired 0-4V output you need to to subtract 1V and  multiply by 2. Fortunately that's easy to do with a single op-amp.



V2 is a reference voltage, you can do that with a divider of reasonably low impedance, or a reference voltage source. If you have a spare op-amp, using that as a buffer for the reference will stabilize things. V1 is the input from your DAC.