First let me say that the phrase does need hyphens, as you typed it -- except that this kind of compound adjective (or adverb) is typed without spaces: "... a check-off-the-list person ..." (Don't be confused by Troi's answer which incorrectly omits the hyphens.) As for the meaning, it derives from the word "checklist", which is simply a list of important actions to be done, or items to be obtained. This question is not completely clear, but seems to be asking if "she" is the kind of person whose enjoyment comes from taking a careful, methodical approach to a task, i.e. creating a checklist and carefully completing each item on the list. The phrase "liking the accomplishment itself" is the part that is somewhat unclear, but I think the "accomplishment" refers to the task of completing the checklist. She is not focused on what the end result is after completing the list. She is just happy to get everything checked off. [Sorry, but I mistakenly posted this answer as a "Comment" first.]

First let me say that the phrase does need hyphens, as you typed it -- except that this kind of compound adjective (or adverb) is typed without spaces: "... a check-off-the-list person ..." (Don't be confused by Troi's answer which incorrectly omits the hyphens.) As for the meaning, it derives from the word "checklist", which is simply a list of important actions to be done, or items to be obtained. This question is not completely clear, but seems to be asking if "she" is the kind of person whose enjoyment comes from taking a careful, methodical approach to a task, i.e. creating a checklist and carefully completing each item on the list. The phrase "liking the accomplishment itself" is the part that is somewhat unclear, but I think the "accomplishment" refers to the task of completing the checklist. She is not focused on what the end result is after completing the list. She is just happy to get everything checked off.

the phrase "check of the list" ultimately means that something was accomplished or has been done.