I overcame a serious illness just after entering college and dealt
with other challenges in the past that didn't make my way easy. I
believe they helped me to build character, but I am not sure if
commenting this might be out of place in a cover letter. My point is
to emphasize that I got what is needed to get over problems in order
to achieve my goals.
You would be telling a "story" that explains why you are such a hard worker and able to overcome obstacles. That could be a compelling story and a great inclusion in your cover letter.
I've read cover letters that mentioned past challenges, military service, and other items that they felt helped shaped their careers. Some of them were very compelling.
Just go lightly on the details of the illness and other challenges. You only want to convey enough to indicate that it was something that needed to be overcome and that you were able to do so. You don't need to include all the details or a long list of illnesses and challenges. If it takes more than 1 or 2 sentences, you probably have gone too far.
And you don't want to include such items as evidence of why you "really need this job". That isn't usually something that an employer will take into account and it could come across as a negative.
Remember, the subject of your cover letter is you. The intent of the cover letter is to convey why you would be a terrific match for this job and this employer. As long as your cover letter focuses on these, adding a small amount of "flavor" is a good thing.
You might wish to have a friend review a draft of your cover letter to give you feedback and make sure you haven't gone over the top.
