The narrator recalls his childhood.
He was forced to work for his own living for Mr. Murdstone's friend Mr. Quinion at London by his stepfather.
He lodges at Mr.Micawber's and goes to his work Murdstone and Grinby's.

Yet I held some station at Murdstone and Gringy's too. Besides that, Mr. Quinion did what a careless man so occupied, and dealing with a thing so anomalous, could, to treat me as one upon from the rest, I never said, to man or boy, how it was that I came to be there, or gave the least indication of being sorry that I was there. That I suffered in secret, and that I suffered exquisitely, no one ever knew but I. How much I suffered, it is, as I have said already, utterly beyond my power to tell. But I kept my own counsel, and I did my work. I knew from the first that If I could not do my work as well as any of the rest, I could not hold myself above slight and contempt.
[David Copperfield by Charles Dickens]
1. I'd like to know if "could" means "as best as he could."
2. I'd like to know "to treat" means "as to treat."
3. I'd like to know if "that" is implied before "I never said" indicating the result of "so" clause.
4. And I'd like to know "How much I suffered" and "utterly beyond my power to tell" are the complements of "it is."
Thank you in advance for your help.
Something has gone wrong with the transcription. It should read "to treat me as one upon a different footing from the rest".

1/2. It basically means "Mr. Quinion did what he could to treat me ..." (which I think you can understand?), but some extra description of Mr Quinlon is included instead of just saying "he".

3. It seems to me that this is anomalous punctuation, and there should be a full stop after "the rest". (I did wonder also whether the comma after "Besides that" could be wrong (or at best misleading) and the structure was supposed to be "Besides that X, Y", an old way of saying "Besides X, Y", meaning that Y happened in addition to X.) In any case, your suggestion is not possible.

4. It is an inverted way of saying "It is utterly beyond my power to tell how much I suffered".
GPYthere should be a full stop after "the rest".
... or a linking "and", of course.