SHould my husband be talking to someone of the oppsite sex that I do not like privately in secret behind my back?
No.
He has lied over and over about talking to her and he uses a form of chat so that I cannot see if he is talking to her.
Double NoNo (lying

)
He tells me that he will not speak to her unless she talks to him or he sees her in public, but yet I find out he is talking to her anyway.
Liars lie.

Assume every syllable he mutters is not true until you can prove/verify it yourself.
He is talking to her about me.
Is "she" married?
If she is, her husband (or her boyfriend) should EVENTUALLY be told about their secret conversations. (LATER, once you have proof)
I do not like this person to begin with, so should he be "friends' with her?
"She" is not your problem.
Your problem is
~~~> your husband does not care if his actions hurt you or cause you emotional distress.
Your HUSBAND is the problem.
My suggestion:
Spy on him secretly.
Put a hidden GPS on his vehicle.
Put a VAR inside his vehicle.
Get ahold of all the bills (credit cards/phone/etc) and carefully scrutinize them.
Read the forum titled "Operation Investigate" for more tips on wayward spouse surveillance tips.
Meanwhile ..... These are your "Do Not" items....
DO NOT1. Make any accusations
2. Make any threats
2. Make any ultimatums
4. Argue
4. Love bust
5. Mention "Her"
6. Beg
7. Cry
8. Pout
9. Show disrespect
10. Let him know you are
suspicious and you are watching his every move. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Once you have done your 'homework' and have hard evidence .... then you come back to this forum and tell us what it is.
We will then give you suggestions of how best to handle what you've found.
YOU MUST BE IN CONTROL OF YOUR EMOTIONS RIGHT NOW !