How can I get a copy of my blood test for drugs/DUI?

All states have rules of criminal procedure that govern discovery.  Usually, an attorney acting on your behalf files a discovery request with the prosecuting agencey.  If you do not have an attorney you can file a motion for discovery.  I have attached the rule governing discovery in the state of California.

1054.1.  The prosecuting attorney shall disclose to the defendant orhis or her attorney all of the following materials and information,if it is in the possession of the prosecuting attorney or if theprosecuting attorney knows it to be in the possession of theinvestigating agencies:   (a) The names and addresses of persons the prosecutor intends tocall as witnesses at trial.   (b) Statements of all defendants.   (c) All relevant real evidence seized or obtained as a part of theinvestigation of the offenses charged.   (d) The existence of a felony conviction of any material witnesswhose credibility is likely to be critical to the outcome of thetrial.   (e) Any exculpatory evidence.   (f) Relevant written or recorded statements of witnesses orreports of the statements of witnesses whom the prosecutor intends tocall at the trial, including any reports or statements of expertsmade in conjunction with the case, including the results of physicalor mental examinations, scientific tests, experiments, or comparisonswhich the prosecutor intends to offer in evidence at the trial.

Hope this helps.