He killed Kate as an act of revenge to get at Paul for killing his brother, Gus Cleary. Paul murdered Gus in the Lassiter's fire a number of years go and kept it a secret. Victor wanted to take someone away that Paul cared about and loved, and make him feel the pain that he felt over losing Gus. For full details, read the below paragraph:
"After emigrating to the UK in 2000, Victor Cleary ended up spending nine years in prison and, upon his release, began looking for the people responsible for the death of his older brother, Gus, who had died in a fire at the Lassiter's Complex in late 2004. He got in contact with Izzy Hoyland, who had once dated Gus when she lived in Erinsborough and was now living in London, and he began to piece together what had happened. After returning to Australia on a false passport, Victor met with his other sibling, Laura, who told him that Paul Robinson had confessed to Gus' murder in 2007, though he'd never been charged as, due to a brain tumour, his behaviour and confession could both have been brought into question. Unable to deal with Paul getting away with killing Gus, an unhinged Victor decided to get revenge and shot Paul's niece, Kate Ramsay, killing her."
