Did Draco Malfoy know that he is related to Sirius Black?

I can't imagine he doesn't know, and he certainly knows after events in the fifth book.

To all wizards, at least the pure-bloods (except oblivious Harry), it's well known that all pure-blood wizarding families are interrelated. When Harry finds out Sirius and Draco are related, Sirius just shrugs it off.

Harry, however, did not laugh; he was too busy staring at the names to the right of Andromeda’s burn mark. A double line of gold embroidery linked Narcissa Black with Lucius Malfoy, and a single vertical gold line from their names led to the name Draco.

“You’re related to the Malfoys!”

“The pure-blood families are all interrelated,” said Sirius. “If you’re only going to let your sons and daughters marry purebloods your choice is very limited, there are hardly any of us left. Molly and I are cousins by marriage and Arthur’s something like my second cousin once removed. But there’s no point looking for them on here — if ever a family was a bunch of blood traitors it’s the Weasleys.”

Sirius is the black sheep in the Black family; they were mostly followers of Voldemort, and some of them even Death Eaters. Both the Blacks and the Malfoys were very old wizarding families, and the fact that Narcissa's maiden name was Black couldn't have escaped Draco for long if he even put the least effort into his own genealogy.

Of course, it is possible that Narcissa Malfoy (née Black) has neglected to mention to her son Draco that she and Sirius Black are first cousins. Sirius was disowned by the Black family, after all, and his portrait burned off the Black family tapestry in Grimmauld Place 12.

Even if she never told him and nobody else did either, Draco would certainly be aware of it after the following interview was published in The Daily Prophet after the mass Azkaban breakout in The Order of the Phoenix:

Speaking to reporters in his private office, Cornelius Fudge, Minister of Magic, confirmed that ten high security prisoners escaped in the early hours of yesterday evening, and that he has already informed the Muggle Prime Minister of the dangerous nature of these individuals.

“We find ourselves, most unfortunately, in the same position we were two and a half years ago when the murderer Sirius Black escaped,” said Fudge last night. “Nor do we think the two breakouts are unrelated. An escape of this magnitude suggests outside help, and we must remember that Black, as the first person ever to break out of Azkaban, would be ideally placed to help others follow in his footsteps. We think it likely that these individuals, who include Black’s cousin, Bellatrix Lestrange, have rallied around Black as their leader. We are, however, doing all we can to round up the criminals and beg the magical community to remain alert and cautious. On no account should any of these individuals be approached.”

It was certainly no secret to Draco that Bellatrix Lestrange was Narcissa's sister, and his aunt. Although she was locked into Azkaban for most of Draco's life, they spent quite an amount of time together after she escaped, and killing her cousin was on top of her to-do list and probably something she bragged about afterwards (she usually called him her "dear cousin" derisively).