A Black Star is generally a Black Sex Link rooster. That does not tell a whole lot, actually. Some hatcheries will cross some of their own breeds to make BSL's. There are many different breeds they can use for the roosters and a few breeds they can use for the hens. A normal combination could be Rhode Island Red rooster over a Barred Rock hen, but there are plenty of other options.
Some hatcheries sell commercial egg layers as Black Stars, or Black Sex Links. These are chickens that have been very selectively bred for many generations to produce highly specialized egg layers suitable for the commercial egg laying operations. These chickens are fairly small and often have a lot of leghorn in them. The small bodies do not take much of feed to maintain large body masses. It is an efficiency thing for commercial egg layers. The ones made by crossing regular breeds are larger and not as efficient on the feed to egg ratio.
That may be more information than you wanted, but chickens tend to have the traits of their parents. If the BSL is from crossing two dual purpose breeds, he will tend to have chicks that are good dual purpose chicks. Of course, that depends on what the hen is too. If he is from the commercial egg layers, his offspring will tend to be smaller and the pullets will probably lay pretty well.
The BSL rooster is a cross. He will almost certainly be black barred in appearance. But genetically, he is split for barred, not pure. That means he has one gene that is barred but one that is not. If the hen he is crossed with is not barred, about half his chicks will be barred and half will not.
That's about all I can tell you with any confidence without knowing more about his parents. Several different genes could be split. For example, it is probable that he is split for extended black. That's not necessary. A Black Australorp rooster over a Cuckoo Maran hen will give you a black sex link that is pure for extended black, but a red rooster will not. This opens up a lot of possibilities. A rooster split for extended black will have about half his offspring black but the other half could be different colors and patterns. And of course, it depends on what hens he is crossed with. If you cross him with a Black Ameraucana, all the offspring from that cross should be black, regardless of his parentage, although half will also be barred. But if you cross him with a hen that is pure for dominant white, you get something else entirely.
Probably not a lot of help. It would help to know somethign about his background and what hens you would cross him with.