Your confusion stems from the use and meaning of the word "character." In literature, "character" refers to an imaginary individual written about in a work of fiction. This character has traits that can be analyzed; emotional, psychological, and/or cognitive growth that can be developed; and attributes and physical qualities that can be described. However, "character" has another meaning and this is the one at play in your assignment. Oxford Advanced Learner's Dictionary defines this use of "character" as being the personal qualities an individual possess. Examples would be how you respond to danger; to praise; to friendship; to pressure etc. This is often referred to as inner character or character traits or even inner character traits. The instructions for your essay are to write your own version of Holmberg's essay in which she is prompted by an offhand remark to examine her inner character and come to some new conclusions about herself and how she understands herself.
What you are asked to do is write under "I believe" or "I doubt" a conclusion or understanding you have come to as a result of some event, remark, influence, epiphany, person's experience, etc. Then you are asked to think of the originating event that led to or precipitated this awakening of knowledge, understanding, or doubt. For comparison, Holmberg tells the event that affected her conclusions and understanding about herself, then tells what her new conclusions and understandings are. She ends with a "brainstorming" list of what might have been the catalyst for the changes in herself: "Maybe it was the impact of my grandfather’s death, or my high expectations, knowledge and experience which is gained with time." In summary, read "Spend a few minutes discerning the values, attitudes, and doubts that may serve to define your character" as "Spend a few minutes discerning the values, attitudes, and doubts that may serve to define your inner character traits."