A: If the children were born here in Chicago and he then left the state, you may file a petition to establish him as the father of the children in Cook County. You will then need to contact a "Process Server" in the city/town in which he lives (use Google), call them, and then fax/scan/mail a copy of the Summons and Petition to Establish the Father Child Relationship.
For the Petition and Summons, look up the court forms on Cook County's website. When you go into the clerk's office to file it, they will certify your Summons and petition, which you will then scan/fax/email to the Process Server you previously contacted.
Give the Process server his name, a physical description, make/model of his car, and any other details that will help him identify the father. The Process Server will then "serve" him with the Summons and Petition you filed with the court, and he will have 30 days to either file his Appearance electronically, or hire local counsel in Chicago to file one for him.
And then.. off you go..!
A: No - the stepmother does not have legal standing to sue the biological mother for child support. In other words, the person attempting to bring the legal action must, in the Court's eyes, have a significant interest or attachment to the subject of the suit/cause of action/claim for child support. Under Illinois Family Law, you must either be the legal guardian, biological parent, or adoptive parent to have sufficient "standing" to bring an action for child support.
A: Yes - you can do it yourself - you would be a "Pro Se" litigant,, meaning you are representing yourself.
If the Child was born in Illinois, and the mother subsequently moved out of state, you may still file in Illinois under the same case you used to established yourself as the Father.
After that, you will need to navigate a rather tricky landscape to have her served with a petition to establish the parameters of your parenting time (visitation). If you came to Chicago from out of state, you will not be able to file suit in Illinois, as the state lacks jurisdiction - you would need to return to the child's home state to resolve the matter.