Assessing a character and the extent to which they are to blame for the outcome of "The Necklace" will depend on your own critical assessment of the story's characters and perhaps also of the story itself. There is a lot of room for interpretation within this assignment and, with it, a wide range of directions in which you can take your answer.
Of this story's characters, the easiest to discuss is Mathilde Loisel herself. She is the story's main character and the character whose personality and motivations are most fully detailed and established within the text. We know that she is driven primarily by vanity and fixated with appearances (particularly with the appearance of wealth and social status), a vanity which shapes the story that follows. The other characters, on the other hand, tend to be more of a mystery, with their inner lives and motivations largely unknown. If you choose to discuss one of those characters, you should be aware of that challenge.
The second part of this assignment revolves around the question of responsibility: Just how responsible is the character for the suffering that befalls the Loisels? Here, too, one can apply nuance. For example, you could easily argue, regarding Mathilde herself, that while she may have been responsible for causing her own self-destruction, this does not mean that she deserved to have her life destroyed. Similarly, you might level some degree of blame on Forestier for her role in having lent Mathilde the necklace to begin with, even while claiming that it was done without malice on her part or while arguing that greater responsibility still falls on Mathilde herself.