I will go by the "standard" attributes of the grim reaper, i.e.: a skeleton, wearing a black cloak, carrying a scythe. While especially the last attribute is quite a bit older, the visual representation of death as the grim reaper most modern Westerners know is primarily rooted in the European Middle Ages.

While the Book of Joel is probably one of the oldest biblical sources use the harvest as a metaphor for death, the most influential is surely the Book of Revelation using the same imagery. Another influnce might be ancient representations of Saturnius (There might even be some interference between biblical and Greco-Roman imagery, but to my knowledge it is nigh impossible to tell what influenced what there). In visual representations, the scythe has been used as an attribute of death since the 13th century in Western Europe (interestingly this is around the same time that people went from using sickles to scythes for mowing).

The cloak can also be traced back to the Middle Ages, when Death was shown wearing a wide variety of "clothing". In the Danse Macabre, Death sometimes wears single items of clothing or is completely clothed like the person he is adressing - a Death figrue approaching a knight might wear the knight's helmet, for example. The item of "clothing" most common - and also most relevant to your question - are cloths or strips of cloth wrapped around the figure. These are most likely burial shrouds, and you can often find the death figures wearing these shrouds draped around their shoulders and over their heads, like hoods. The "modern" cloak is most likely derived from these depictions of death as a skeleton wearing (parts of) a burial shroud as a makeshift cloak.

I hope this answers your question. :-)

Sources

Ariès, Philippe. Western Attitudes Toward Death from the Middle Ages to the Present. John Hopkins UP, 1974.

Warda, Susanne. Memento Mori: Bild Und Text in Totentänzen Des Spätmittelalters Und Der Frühen Neuzeit. Böhlau, 2011.