Sirius himself says at some point in Goblet of Fire that Crouch pushed through legislation to have him convicted and thrown in Azkaban without a trial.
There seems to be a general consensus, specifically voiced by Fudge but also implicit in Crouch pushing through this legislation, that Sirius was considered one of Voldemort’s innermost, most dangerous, and most trusted right hands. Easily as inner-circle as the Lestranges, for example.
It seems quite unlikely to me that someone so close to Voldemort should not be a Death Eater—apart from Fenrir Greyback (who I would think is much less inner-circle), I do not recall hearing of any of Voldemort’s important servants not being Death Eaters.
As far as canon describes, all Death Eaters have the Dark Mark magic-tattooed on their arm, to be used as a way for Voldemort to summon them to him. Considering that Karkaroff, who had definitely burnt all his Death Eater bridges, still had the Dark Mark, I am assuming that getting rid of the Mark is not possible: once it’s on, it stays on.
Yet, as Hermione points out in Order of the Phoenix (ch. 16, “In the Hog’s Head”), Sirius does not have the Dark Mark.
So why was he so swiftly convicted and locked up? I mean, I know they were eager to be seen to be doing something (as Fudge keeps pointing out is his excuse for most of the miscarriages of justice he commits)… but seriously, did no one think to check if Sirius actually had a Dark Mark on his arm or not? Did the Ministry of Magic not even know at the time that all Death Eaters had the Mark? Or were they just happy to accept—without the scrutiny of a full trial—that a non-Death Eater would commit mass murder for a Voldemort who was presumed defeated?