The "dispute" between Michael and the Devil over Moses' body comes from the Book of Enoch or an earlier work called "The Assumption of Moses".
From this source:
Aside from Jude 9, there is no biblical record of any “contention” or meeting between the devil and Michael the archangel. Many scholars, based on the writings of Clement, Justin Martyr, Irenaeus, Origin, and Didymus (Guthrie, 1962, p. 918; Earle, Blaney, and Hanson, 1955, p. 411), assume that Jude 9 is a reference to an apocryphal book called The Assumption of Moses, a work that is extant only in fragmental form (in Latin and in a translation from Greek). The fragment now known as The Assumption of Moses presents the account of Moses’ appointing of Joshua as his successor, and a description of the future of Israel during the conquest of the Promised Land. According to Richard Lenksi, scholars believe that the missing portion of The Assumption included “an elaboration” of Deuteronomy 34:5, the biblical account of Moses’ death, showing how God used angels to bury Moses (1966, pp. 601-602). It is thought that The Assumption of Moses, at this point, used Zechariah 3:1-2 as its basis for the use of the phrase “The Lord rebuke you!” It has not been proven, however, that Jude intended to quote from The Assumption of Moses.
Now, if you'll permit a bit of personal exegesis - The word "disputing" comes from the Greek - diakrinomenos - which means to "to take issue." The full text is:
tw diabolw diakrinomenoV dielegeto
The root of that word - krinimos, means to make an accusation against. That it is followed by dielegeto (to converse, discuss) seems to indicate that this "dispute" is more of a claim / counter-claim thing than a fight.
Basically, the devil is saying - "Hey, Moses' body belongs to me, not you!" He need not be in Hell to make a claim to the body. Indeed, as Deuteronomy 34 says, the location would have been on Mt. Nebo in Moab - not Hell!:
5 And Moses the servant of the Lord died there in Moab, as the Lord had said. 6 He buried him[a] in Moab, in the valley opposite Beth Peor, but to this day no one knows where his grave is. 7 Moses was a hundred and twenty years old when he died, yet his eyes were not weak nor his strength gone. 8 The Israelites grieved for Moses in the plains of Moab thirty days, until the time of weeping and mourning was over.
Note also that it is the body and not the soul. Since at least some debate whether or not the ancient Jews had the concept of an afterlife, it may be fair to characterize this as a battle in which eternal death and resurrection are competing, in a sense, for Moses' remains.
In any event, we know the outcome from a much more accepted source, however. During the Transfiguration, Moses appears, along with Elijah. Elijah had been "caught up into heaven" without dying, and it may be that Moses did the same. Regardless, the Devil didn't win here, since Moses was present.