How does Parkinson's disease result in tremors? From what I understand, Parkinson's disease is caused by the death of dopamine producing cells in the substantia nigra, however I don't understand how that causses the symptoms of Parkinson's. I am particularly interested in the tremor aspect of the disease, although it would be great if someone could also explain how the other psychiatric problems arise (dementia, emotional problems and sleep issues).

The reason for this question is that I am trying to understand how deep brain stimulation (DBS) is able to stop the tremours in Parkinson's, or at least reduce their severity, and whether DBS could also help in the other symptoms of Parkinson's.

What causes the tremors in Parkinson's disease? Why do Parkinson's patients develop dementia, emotional problems and sleep issues?

Why does Deep Brain Stimulation alleviate tremors in Parkinson's disease? Can DBS help other Parkinson's symptoms?
Parkinson's Disease, as you note, is believed to occur due to damage or degeneration to the basal ganglia, especially the substantia nigra. Cells in the basal ganglia and substantia nigra produce dopamine, and in fact, the majority of the brain's dopaminergic cells have their bodies here, but that isn't the whole story. From Kandel's Principles of Neural Science (the first rule of neuroscience is always to look in your PoNS):

Primarily on the basis of studies in MPTP-treated primates, a working model of the pathophysiology of Parkinson disease has been
developed. According to this model, loss of dopaminergic input from the substantia nigra pars compacta to the striatum leads to increased
activity in the indirect pathway and decreased activity in the direct pathway (see Figure 43-6) because of the different actions of dopamine on
the two pathways (via D1 and D2 receptors, respectively). Both of these changes lead to increased activity in the internal pallidal segment, which
results in increased inhibition of thalamocortical and midbrain tegmental neurons and thus the hypokinetic features of the disease.

Tremors and DBS
Three of the central nervous system's four major dopaminergic pathways originate in the substantia nigra, but it is the nigrostriatal pathway which is believed to be involved in Parkinson's tremors. The nigrostriatal pathway innervates the striatum, which is involved in planning movements. Broadly speaking, because the basal ganglia are involved in action selection and routing information, impaired communication along this pathway can explain tremors as a function of failing to inhibit undesirable actions and facilitating desirable actions (Stocco, Lebiere and Anderson, 2010).
Deep Brain Stimulation (DBS) has been shown to be effective in alleviating the symptoms of Parkinson's Disease in a number of studies (e.g., Deep-Brain Stimulation for Parkinson's Disease Study Group, 2001; Weaver et al., 2009). However, comparative studies frequently do not explain the theoretical grounding for the method. Kandel provides the answer (first rule of neuroscience):

Selective inactivation of the sensorimotorportion of either the subthalamic nucleus or the internal pallidal segment is sufficient to ameliorate the cardinal parkinsonian motor signs (akinesia, tremor, and rigidity) in MPTP-treated animals (Figure 43-7). Surgical lesions of the posterior (sensorimotor) portion of the internal pallidal segment (pallidotomy) in patients with advanced, medically intractable cases of Parkinson disease is also highly effective in reversing parkinsonian signs. [...] Thus the hypokinetic features of Parkinson disease appear to result from increased (inhibitory) output from the internal pallidal segment as a result of increased (excitatory) drive from the subthalamic nucleus.

DBS treatments for Parkinson's take advantage of this finding to achieve their effect by directly stimulating the subthamamic nucleus and/or globus pallidus. This implies that the treatment is extremely specific to the mechanism, and though there is no reason to rule out effectiveness in other areas, there is no reason to rule it in, either.
Emotional problems
Parkinson's Disease patients are, perhaps intuitively enough, prone to depression, and as many as 25-40% of sufferers experience depression (Leentjens, 2004). Less intuitive is the finding that future patients are at increased risk of depression in the three years preceding their Parkinson's diagnosis (Leentjens, van den Akker, Metsemakers, Lousberg et al., 2003). One explanation for this finding, however, may lie with the other two dopaminergic pathways in the substantia nigra (mesolimbic and mesocortical), which are involved in affect and emotion, and that the emotional problems arise from the dopaminergic impairment in a manner similar to the tremor symptoms.
Unfortunately, I do not know a mechanism for dementia or sleep issues.
References

Deep-Brain Stimulation for Parkinson's Disease Study Group (2001). Deep-brain stimulation of the subthalamic nucleus or the pars interna of the globus pallidus in Parkinson's disease. The New England journal of medicine, 345(13), 956.
Kandel, E. R., Schwartz, J. H., &amp; Jessell, T. M. (Eds.). (2000). Principles of neural science (Vol. 4, pp. 863). New York: McGraw-Hill.
Leentjens, A. F. G. (2004). Depression in Parkinson's disease: conceptual issues and clinical challenges. Journal of Geriatric Psychiatry and Neurology, 17, 120-126.
Leentjens, A. F. G., van den Akker, M., Metsemakers, J. F. M., Lousberg, R. &amp; Verhey, F. R. J. (2003). Higher incidence of depression preceding the onset of Parkinson's disease: A register study. Movement Disorders, 18(4), 414-418.
Stocco, A., Lebiere, C., &amp; Anderson, J. R. (2010). Conditional routing of information to the cortex: A model of the basal ganglia’s role in cognitive coordination. Psychological review, 117(2), 541.
Weaver, F. M., Follett, K., Stern, M., Hur, K., Harris, C., Marks, W. J., ... &amp; CSP 468 Study Group. (2009). Bilateral deep brain stimulation vs best medical therapy for patients with advanced Parkinson disease: a randomized controlled trial. Jama, 301(1), 63-73.