Pepe joined the Spanish champions from Porto, with whom he won two Portuguese titles and one Portuguese Cup.
He had offers from other European clubs but opted to move to Madrid.
He told the club's official website: "Everyone knows I had other offers, but my dream was to join Real Madrid.
"This dream has now come true. I want to be very successful here."
Pepe will have to compete for a starting spot with veteran Italy international Fabio Cannavaro and Germany centre-back Christoph Metzelder, who also joined the club this week.
But he denies he will be third choice in the pecking order.
"I don't consider myself as being the team's 'third' centre back. I'm going to work very hard and I am always going to be a humble player," he said.
"It will be up to the coach to decide what to do, but I promise you all that I am going to sweat in this shirt."
Pepe is set to make his league debut on the opening day of the season when Madrid face city rivals Atletico.
It is a match he is already looking forward to.
He said: "I know it's a derby and derbies are always important.
"Making my league debut in such a game is the best thing that could happen to me. I find such pressure very stimulating."
