Is there a bias? If so, what is the nature in the story?

Sherman Alexie's story is funny and relevant to our times. You asked about the possible types of biases in the story. If you are looking for an overarching bias, I would argue that there is an implicit racial bias in Alexie's story. The word "implicit" tells us that this particular type of bias is inherent in nature. If we have this type of bias, we stereotype others unconsciously. It is not something we purposefully think about.
All of us potentially have implicit racial biases, despite our best intentions otherwise. In the story, Jackson Jackson displays all the behaviors people commonly associate with Indians who are jobless and homeless. For example, he is an alcoholic and keeps bad company. Additionally, Jackson does not have any ambitions in life, often makes poor financial decisions, and enjoys telling fibs. Aside from the racial bias, we have what we call the "Horn Effect." This is similar to racial bias.
The Horn Effect occurs when we allow what is considered a negative trait in someone to prejudice our subsequent interactions with them. For example, during Jackson's interaction with the pawnbroker, Rose of Sharon interjects. She tells the pawnbroker that they could go to the cops if he does not return the regalia that belonged to Jackson's grandmother. The pawnbroker answers: “Well, I suppose you could go to the cops,” he said. “But I don’t think they’d believe a word you said.”
Here, we have an offensive racial bias: Indians are always telling fibs/stories. Therefore, no one should believe a word they say. Then, we have the Horn Effect. This is shown when cops are believed to be inherently biased against Indians. Additionally, when an action performed by an officer is considered biased, we judge all his subsequent actions accordingly. In his response, the pawnbroker unconsciously demonstrates both a racial bias and the Horn Effect. Alexie turns both biases on their head in his interesting story.
First, Jackson is shown to have a resilient spirit. He is always cheerful despite his trying circumstances. Additionally, he does not harbor bitterness or anger toward anyone; he is invariably kind to others. Alexie also suggests that Jackson's love of stories comes from a long tradition of storytelling by his people. Thus, storytelling should not be confused with the practice of telling lies.
Later, Jackson's interaction with Officer Williams shows that good cops exist. Before the two part ways, Officer Williams gives Jackson thirty dollars; it is the only money he has until payday. He offers it to Jackson because he believes in Jackson's dream.
Yes, there are biases in the story. However, Alexie shows us that we can prevent them from defining our interactions with others. It is a powerful story.