I have no experience in this but, I have been thinking about adding a chat service to my website. So, what I am about to share is just really what I believe would be the process.

First, here is a website that tells you a little about how to charge for the chat service. help.instantgo.com

And on the other hand, I believe it is more of a consultation service that you are providing and people do this all the time. I just never seen anyone do it through chat, although, I know people do it. I just never came across anyone yet.

Anyhow, so, what I have seen people do is they would personally set up a Paypal business account.

Find the chat service that you like. and then charge people for your consultation time and they pay you through PayPal.

Also, you would have to have it set up some kind of way to where they pay you first then, receive your services.

I would try and ask a chat service the question and see if they give you any responses or ideas on how to go about this.

I am sorry that I was not too much help but, I did my best in giving you some ideas.