Investigation was carried out by judges appointed by the Emperor to oversee county-sized administrative divisions. The judge would both investigate the crime, examine the evidence, charge the accused, interrogate them, and sentence them based on the Code of Punishments promulgated by the Emperor.

Interrogation involved the judge's use of the 'Five Listenings'; they would gauge the subject's statements, expression, breathing, reaction to the words of the judge, and eyes. Traditional Chinese Law only allowed criminals to be sentenced upon having confessed, so with the permission of his superiors, the judge could resort to interrogative torture; a light, whippy stick less than a quarter of an inch thick would be used to strike them up to two hundred times in three sessions twenty days apart. If the subject of interrogation had been accused and had not confessed after torture, the accuser would be interrogated in the same manner. If they confessed to a false accusation, they would suffer the same punishment as the accused would have if they were sentenced.

There were five traditional punishments in Chinese law: whipping with a light stick, caning with a heavy stick, penal labor, exile, and execution. Furthermore, each punishment was graduated into degrees; while the punishment for a given crime might be qualitatively the same, specific aspects such as the relation of the perpetrator and the victim could increase or decease the degree,

In your case, you should also be aware of the Ten Abominations. Imperial China was a very heirarchical place, and there were many people who enjoyed reduced punishments in a system called the Eight Deliberations. Investigating and sentencing relatives of the emperor, meritorious officials, and so on required the approval of the Emperor. This is where the Ten Abominations come in. These were crimes so heinous that all legal privileges were suspended for the investigation and sentencing. Plotting treason and rebellion were at the top of the list, carrying with them the penalty of one's entire family being executed, but also abominable was the murder of one's parents, of government officials, and of three or more innocent people of one family. The punishment for killing three or more people of a family (or dismembering bodies after death) was death by slow slicing; any wives of the accused would be exiled ~500 miles, and all property would be given to the victims' closest descendants.

Followup: If someone knowledgeable in the topic is familiar with Robert Van Gulik's Judge Dee novels, how accurate are they in portraying Tang Dynasty China? Are there any significant historical discrepancies?

County and prefectural magistrates usually dealt with criminal cases within their jurisdiction, but if the case was severe enough, the central government might step in. The central government organ of investigation was the Dali Si (大理寺), who investigated cases on behalf of the emperor.