Testosterone and dopamine are intimately connected in a bidirectional relationship. Testosterone regulates dopamine and dopamine testosterone. Because many of the men that show up to my site are low testosterone, I want to start with the research that shows that testosterone profoundly affects brain dopamine levels. Testosterone profoundly affects many other neurotransmitters as well, which I over in my link on Testosterone and the Brain.
Here are three keys ways that testosterone and dopamine influence each other:
1. MPOA (Medial Preoptic Area). This region of the brain sits near the hippocampus and is critical for the bedroom. If this area is injured, animal studies show that no copulation will occur. And one study in rats showed this by microinjecting dopamine stimulators into the MPOA, which increased mountings, and dopamine suppressors, which did the opposite.  Therefore, it is obviously dopamine in this region that is one of the key gatekeepers.
What testosterone have to do with this? Research, again in animals, has shown that testosterone controls dopamine release in the MPOA. In fact, castrated animals show no interest in mounting and had no dopamine release in the MPOA. Castrated animals given testosterone, however, would do their duty with a female AND had the necessary dopamine release.  The clear takeaway: testosterone governs dopamine release in key areas of the brain.
2. Low Testosterone Can Destroy Dopamine. So testosterone can stimulate release of brain dopamine and low testosterone can inhibit it. However, preliminary evidence shows that low testosterone can have a much more nefarious side and may play a role in Parkinson's Disease. Of course, Parkinson's is the well-known condition that afflicts over a million people and is characterized by muscle rigidity and tremors due to loss of brain dopamine levels.
What researchers at Rush University Medical School found out is that mice who suddenly lost their testosterone had a rapid rise in iNOS levels. Many of you are familiar with eNOS, which is the endothelial nitric oxide synthase that Viagra and Cialis act on and which dilates your arteries. iNOS, inducible nitric oxide synthase, is its less well-known cousin and is involved in immune and inflammation response.
iNOS is known for releasing large amounts of nitric oxide as a defensive tactic. Nitric oxide is a free radical that can be used as a weapon and, therefore, sudden low testosterone essentially puts a huge oxidative load on the brain and may actually do some damage that leads to Parkinson's.  A further verification of this is the fact that, according to one study, men are 1.5 times more likely to contract Parkinson's and that low testosterone levels have been associated with Parkinson's.  In addition, giving male Parkinson's patients testosterone can help with tremors. 
NOTE: It is important to point out that there have been other key factors identified in the progression of Parkinson's, including copper and pesticides. I cover the latter in my link on Pesticides and Parkinson's.
3. Dopamine Stimulates Testosterone. As you know from my link on Testosterone and Your Mind, competition and the brain's reward system can yield very significant increases in testosterone. And, of course, front and center to the brain's reward system is dopamine. Researchers have found that "the present data clearly demonstrate that GnRH mRNA levels are positively regulated by dopamine." 
It is also VERY important to note that testosterone's impact on the male brain is extraordinarily profound. I know that when I got my first testosterone cypionate injections, I felt like "fireworks" were going off in my brain. Low testosterone is linked with depression, mood disturbances, fatiuge, anxiety, decreased working memory, mental fog and on and on the list goes.
Yes, testosterone is the Bedroom Hormone. Yes, testosterone is the Muscle Hormone. But, more than that, testosterone is the Brain Hormone.
1) Physiology and Behavior, 2005, 86:356-368, "Dopamine, the medial preoptic area, and male s_xual behavior"
2) Hormones and Behavior, May 2001, 39(3):216-224, "Testosterone Restoration of Copulatory Behavior Correlates with Medial Preoptic Dopamine Release in Castrated Male Rats"
3) J Neurol Neurosurg Psychiatry, 2004 Apr;75(4):637-9, "Are men at greater risk for Parkinson's disease than women?"
4) http://www.rush.edu/webapps/MEDREL/servlet/NewsRelease?id=1712, Rush University Medical Center press release, July 26, 2013, "Sudden Decline in Testosterone May Cause Parkinsonï¿½s Disease Symptoms in Men"
5) Arch Neurol, 2002, 59(5):807-811, "Refractory Nonmotor Symptoms in Male Patients With Parkinson Disease Due to Testosterone Deficiency"
6) Journal of Clinical Neuroscience, Jan 2006, 13(1):133ï¿½136, "Testosterone improves motor function in Parkinsonï¿½s disease"
7) Endocrinology, 1992 Jul, 131(1):395-9, "Role of dopamine in the regulation of gonadotropin-releasing hormone in the male rat brain as studied by in situ hybridization"