I am including a link below to the Rules of the Supreme Court. Rule 13 provides a 90 day time limit for filing a petition for a writ of certiorari. Rule 30 excludes the day of the event from which relief is being sought, so you do NOT count the date of the appeals court decision. Also you include all intervening Saturdays, Sundays and legal holidays, except where the last day of the 90 day period FALLS on one of them.
Now, on a more serious point:
You need legal counsel to have your best chance of review at the Supreme Court. But even with legal counsel, the chances of any particular petition being granted are exceedingly small. Read Rule 10 and give some thought to the points made there: the Court's principal task is bringing uniformity to federal decisional law and uniformity to interpretation and application of federal laws, regulations and the US Constitution.
This answer is not a substitute for consulting with and retaining the services of an attorney for your legal needs. By providing this answer, I am not entering into an attorney client relationship with you.
Are you saying that you were litigating pro se at one of the U.S. Court of Appeal and lost? If so, you might first consider seeking rehearing or rehearing en banc. Though those are difficult paths, taking a pro se employment case to the U.S. S. Ct. is an extremely long shot. You should consult with counsel at this point as it sounds like time and options may be running out on your case.
This is not formal legal advice.