What rule or rules of law did Chief Justice Taney rely upon when making his decision?

The most famous case that Taney ruled on was the Dred Scott v. Sandford case in 1857; in answering your question, I will reference that case. Scott sued for his freedom in Missouri after his master took him to army postings in Illinois, Wisconsin, and the Minnesota Territory. According to the Northwest Ordinance of 1787, these areas were supposed to be free. Also, the Missouri Compromise of 1820 ruled that slavery above the southern border of Missouri (Missouri excepted) was illegal. Scott sued for his freedom in Missouri based on the precedents that if one lives in a free territory, one therefore is free. The trial court freed Scott, but upon review by the Missouri Supreme Court, Scott was returned to slavery. Scott appealed his case to the U.S. Supreme Court.
Writing for a 7-2 majority that ruled that Scott was still a slave, Chief Justice Roger Taney, a slave owner from Maryland, stated that Scott, like free blacks in the North, was not a U.S. citizen and he was barred from suing in court. He also ruled that the Compromise of 1820 was unconstitutional as it placed limits on one's personal property rights. He also claimed that neither Congress nor territorial governments could ban slavery. Popular sovereignty, a hot issue at the time, was therefore rendered moot in the eyes of the court as a territorial government could not legislate on slavery. Also, black people of any social class had no legal rights in the United States.
This case would go on to become one of the tipping points which led to the Civil War. Abolitionists were outraged that a slave-holding judge was able to rule on a slavery case. Some Southerners viewed the case as the Supreme Court defending their right to their slaves.
https://www.blackpast.org/african-american-history/dred-scott-v-sandford-1857/