I'm curious to see if I am eligible for British citizenship. My brothers and I were born in the US between 1957 and 1961 to an English mother (she was born in England) and a British father (he was naturalized at an early age). Both parents are now deceased. I still do have Aunts and Uncles as well as most of my cousins living in England and Wales (although I doubt this has any bearing).
This is one of the cases where the government's citizenship checker is not really helpful, but let's try to answer the questions newertheless:
When were you born?
Before 1 January 1983
As you were born before 1983
Were you born in the UK or a qualifying territory?
No, the US is not one of the qualifying territories
Were you a UK and Colonies citizen on 31 December 1982 with right of abode in the UK?
Now this is the tricky question. This basically asks you whether you are a British citizen or not, so to answer it you need to check the section 5 of the 1948 nationality act. Based on this you are already a British citizen by descent, if your father was a British citizen not by descent before you were born.
You've said that your father has been naturalized as British citizen, so he is a British citizen not by descent. If he had done this before you were born, then you are already a British citizen by descent.
See my other answer on how to apply for your first British passport from abroad.
If your father naturalized after you have been born, then you are not a British citizen, as mothers could not pass on citizenship before 1983. However, you can try to register as British citizen. To register you need to have all of the following:
You were born before 1 January 1983
This is okay
You would have become a citizen of the United Kingdom and Colonies by descent if women had been able to pass on citizenship to their children in the same way as men at the time of your birth
This is also okay, as your mother was a British citizen not by descent when you were born
You have right of abode
You have multiple ways where you can get this, fortunately as
your mother was, at the time of your birth, a citizen of the United Kingdom and Colonies by birth, legal adoption, naturalisation or registration in the United Kingdom, Channel Islands or Isle of Man
applies to you, this part is okay. And finally
the Secretary of State is satisfied that you are of good character.
if you haven't commited a crime, act of terrorism, and you haven't tried to illegally enter the UK previously, etc. then this should be okay as well.
If all of this aplies you have to register for British citizenship using form UKM. Fortunately you only need to pay the ceremony, which is £80 as of 2015 per adult. You will also need two referees to sign the form, which has it's own rules. You should defintiely read the form UKM guides and maybe also hire a solicitor to help you with it to make sure everything is in order. The guide will also contain where to send the application form and related documents (like your current passport), which is as of 2015:
Department 1 UK Visas and Immigration PO BOX 306 Liverpool L2 0QN