This controversial work is purportedly a translation of a sequence of pictographs which give the epic of the Delawares, a tribe which lived in the central Eastern seaboard. Taken at face value, this would be one of the few actual written texts from Native North America, including a clear account of an eastward migration over the 'stone-hard water'. The source of the document, as well as aspects of the Delaware text, and some of the historical episodes have been called into question. I'm not going to rehash this discussion here, but offer some comments based on the content of the text.
Most likely, the Walam Olum was forged in the 19th Century by someone who was attempting to provide a mythological underpinning for the theory that Native Americans migrated from Asia at some point in the recent past. It is now believed that this migration took place between ten and fifteen thousand years ago. Instead of a mass emigration over a frozen ocean, it was a gradual infiltration by small groups. They weren't out to discover a new world, but simply following their food sources. Initially they moved over a land bridge that connected Asia and America (Behringia), which was exposed at the time by the the greatly lowered Ice Age sea levels. When the glaciers contracted at the end of the Ice Age, the conventional theory is that a 'corridor' through western Canada was created, which served as a migration route south into North America. Another theory is that people could have taken a sea route along the Canadian coast to bypass the ice sheets. In any case, the archeological record indicates that it took hundreds or thousands of years for people to get from Siberia to the shores of the Atlantic.
Not only does seems unlikely that any record of this migration would have been preserved to this day (let alone in writing), but this account is at odds with most other Native American origin stories. The typical Native American mythology assumes that 'the people' have always lived here, or emerged from one or more worlds underneath the earth. For this reason, many traditionalist Native Americans regard the Asian land bridge migration theory in the same way that fundamentalist Christians feel about Darwinism.
Other aspects of the narrative mark this as a contaminated text. The Walam Olum origin myth involves an act of creation of all things by a 'Manitou', who subsequently battles an evil 'Magician' who brings death, disease and bad weather into the world. A genesis at the hands of a single masculine deity, and an ensuing cosmic struggle between good and evil, so central to Old World mythology, are, simply put, foreign concepts in Native American mythology.
For instance, in many Native American cultures, the world is created by an animal who dives to the bottom of the ocean and brings up land bit by bit, or by a pantheon of fathers and mothers. There is usually a trickster figure (in the Southwest, Coyote) who commits transgressive acts. However, the trickster is not considered 'evil' in the Judeo-Christian sense, but comic or stupid. Death, disease and so on are often brought into the world through an accident or misunderstanding, not as an intentional punishment by some entity. Another aspect of Native American mythology is the fluid boundary between the animal and human worlds, of which there is no evidence in the Walum Olum.
There may be some actual indigenous content in this text, but in retrospect this is either a unique outcropping of Old World mythology in an improbable location, or a 19th Century forgery. There are some similar and better documented pictographic records from the Plains area, but these have nowhere near the epic scope of the Walum Olum. In any case, this remains one of the lasting mysteries of Native American literature.