How can I sue my boyfriend who is a citizen of the United states of American for child support and I am a Kenyan citizen?

Firstly you need to establish positive paternity with a DNA test.
This will most likely need court action.
The rest is easy.

The Kenyan constitution adopted in 2010 provides for children's rights including the obligation for both parents whether married or not, to support their children. Children's Rights and the new Constitution in Kenya (http://www.aclr.info/index.php?option=com_content&task=view&id=540&Itemid=10)

So, you need to contact your local equivalent of Family Court for what the process is to obtain a support order. The problem you will have is if your boyfriend is no longer in Kenya. If he is, then you may be able to put a hold on his passport restricting him from leaving the country. At least until a paternity test can be done.While the US is party to many treaties for reciprocal treatment of child support, I don't know if there is a treaty with Kenya, so you may have problems compelling him to honor and order issued by a Kenyan court.

This site lists attorneys in Nairobi specializing in this branch of law:
Nairobi, Kenya Child Support and Custody Law Firms, Lawyers | Nairobi, Kenya Law Firm (http://www.hg.org/law-firms/Child-Support-and-Custody/Kenya/Nairobi.html)