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
“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).