I am totally confused if I need to do anything with my Biometrics permit or can I just renew the USA Passport and keep the expired Passport with my Biometrics permit that shows my ex-married name?
The name on your new US passport MUST match the name on your BRP.
So, if you renew your passport in your new married name, then you have 3 months in which to also change your name on your BRP, or you could be fined £1,000.
Or, if you renew your passport in your ex-married name, then you don't need to do anything with your BRP and you can just keep everything as it has been for the last few years.
If you do leave your passport and BRP in your ex-married name, then you must book all travel in that name too... because the name on your plane ticket must match your passport and BRP exactly. So, you don't carry your marriage certificate when you travel, you just book everything in the ex-married name.
So it's up to you which you would prefer to do. To be honest, I'd advise renewing your passport in the ex-married name so you don't have to worry about the BRP. There are many people here on UK-Y who have their passport and BRP in their maiden name and use their married name for everything else in their lives.
If you decide to apply for UK citizenship (which you qualified for in April 2014) you can just get your UK passport issued in the new married name. At this point your BRP will be cancelled, so once that has happened, you can change your name on your US passport without having to do anything else.