What exactly needs to be included when "writing up" an employee? Enample for being late all the time.

Reference to documentation such as something showing the employee's acknowledgement of what the specified hours of work are. Time sheets or "clock-in" records if applicable. Pay records to show if hours docked from pay because of lateness. Not necessarily including any of the documents with the reprimand (or whatever "writing up" means in your organization) but each reference to documents or rules will help support your actions.
Also, were there reprimands or counseling sessions previously warning the employee of penalties for lateness?
If the employee has a habit of being late all the time there would most likely be some written records or a reference to oral discussions.
And all depends on the rules for your particular worksite. Are unions involved?
Civil service regulations? Have you discussed your plan with your supervisor?
Are there documents or can you show that show low production, missed meetings, complaints from others etc.
All of the above will be helpful if this ends up going further. Does your employer have a policy of progressive discipline? Only you know what your employer's policies are and how big a deal this will be.
Do you have the supervisor's OK to proceed?