Can I get my own blood alcohol test following a DUI arrest?

Yes, but the test must be drawn by an agency qualified to do so in order for the evidence to be admissible in court, and the test must be done immediately upon your release from police custody to have any potential value to your defense. Frankly, having your own test done is rarely beneficial because of the difficulty in getting it done quickly enough relative to the time of the arrest and by a qualified agency.