This is a tough one. I think it's difficult because there has to be a distinction made between "racism" and "issues of race" in the drama. I am not sure if one is able to claim that Shakespeare is demonstrating racism. It might be more of a limitation in suggesting that Shakespeare is mirroring the racial attitudes of his time, and not as much "racism." Adding to this is the real issue that Shakespeare might be critiquing his time frame's attitude towards race, and his use of racial language is a reflection of this. With Shakespeare, there is always complexity and this is no exception. The discussion of racism in this light is difficult to assess.
However, I think that a clear case can be made to suggest that Shakespeare is able to construct the character of Othello who is marginalized and a figure who is riddled with self- doubt for a variety of reasons, of which race is one factor. The idea of "insider vs. outsider" is crucial to how Othello sees himself and how others see him, and even more vital in his downfall as the latter begins to infect the former. In this setting, "the other" as racial identity plays a role. As the first moments of doubt and insecurity begin to creep into his psyche, it is not accidental that Othello cries, "Haply for I am black." It is relevant that skin color enters his frame of reference as his world starts to disintegrate. Shakespeare draws the idea that Othello is different from the rest of Venetian society because he is black and represents "the other." This becomes one of the reasons that he is unable to fully embrace his status as an "insider" and one of the reasons why he is unable to fully accept the idea that Desdemona would be in love with him:
It is that he is, from the beginning, placed in a position of isolation from the other characters. In the same way, Hamlet's black clothes isolate him visually from the rest of the Danish court. This isolation is such an integral part of Othello's experience that it is constantly operative, even if not necessarily at a conscious level. Anyone who is black would appreciate its importance in understanding the character of Othello. Before he appears, our attention is forcibly focussed on Othello's race. The speeches of Iago and Roderigo in the first scene are full of racial antipathy. Othello is ‘the thick-lips’, ‘an old black ram’, ‘a lascivious Moor’ and ‘a Barbary horse’, and he ‘is making the beast with two backs’ with Desdemona. The language is purposely offensive and sexually coarse, and the animal images convey, as such images always do, the idea of someone who is less than human.
Racial issues are present in the play. Shakespeare uses them to enhance his construction of a character that might hope to be an insider, but is condemned to be an outsider, one who wishes to be at the center but is forever marginalized by a fatal cocktail of his own devices and social construction. Yet, the idea of "racism" in the modern sense has to be reevaluated. I would also examine the many essays that enotes has on the subject, as only one is linked below.