How is DNA used to Check for genetic relationships across time?

When a skeleton was found under a car park in Leicester, how was it proved beyond reasonable doubt to be that of King Richard III?

Fortunately, Leicester University was equipped to carry out a range of scientific tests to do just this.

You can look up information about the structure of DNA in any A level textbook.

What A level does not tell you much about is mitochondrial DNA (mtDNA) which is inherited only from mothers. This means that whilst a male will have the same mtDNA as his mother, he cannot pass this on to his own children. 

However, an unbroken line of female descendants will carry the same mtDNA.

So the challenge with the skeleton was to find living descendants of female relatives of Richard III tracing back over 500 years. Remarkably this was achieved by Dr John Ashdown-Hill and is explained in this link

You can read more about mtDNA in Nature Education