Send back the necessary papers to appeal the claim. All your request for an appeal needs to say is, "I wish to appeal this decision." Be sure you submit it within the time limit for an appeal. You may have the option of requesting a telephone or an in person hearing. If so, I usually think it's a better idea for the claimant to request an in person hearing, as this is much less convenient for the employer and if they do not have a very good case, sometimes they don't even show up. You, will of course still have to appear and present your information to the hearing officer.
Keep making your weekly certifications during this time, whether or not you are receiving any benefits for these weeks. If you are approved, you will be back paid for weeks you have certified for, none for weeks you haven't.
Tell your situation as you have presented it here. Well, not exactly. Why do you say that the reason you got fired was the incident where you asked the guy if he was a rapist at Halloween? Did the employer tell you this? If so, what he said needs to be repeated, not anything you may have assumed was the reason you were fired. What exactly did he say when he told you you were fired? If he wrote you a letter three weeks afterward and said you had quit....was this three weeks after he had fired you, or three weeks after this incident and you had been working up till then?
Did you ever, at any time in the interactions with your boss, say you quit? (As in something like, "Well, you can't fire me, you jerk, cause I quit!") I hope not, for the sake of your being approved. If you quit a job, you must prove you had a very good job related reason to quit. But it doesn't sound as though that is what happened here.
You need to make this very clear to the employment office appeals hearing officer. You did not want to quit your job, you did your job to the best of your abilities, you did not do anything knowingly to cause yourself to be fired. You had had no prior warnings or indications that anything you were doing was about to cause you to be fired. If you did anything wrong as far as mentioning the word "rapist" to your co worker while off duty, this certainly should not be classified as gross misconduct, unless there was a whole lot more inappropriate verbage included in this incident that we're not hearing about.