I have some chained methods in my experiment like A→B→C . What is the best of way of expressing putting something new on this chain? Suppose that I want to add D to have A→B→C→D. Do I say I add D on the chain? into the chain? to the chain? etc. Any suggestions are appreciated.
In OP's specific case, D is being added to [the end of] the chain, so "to" is fine. But if the final sequence is A → B → D → C, then D was added (inserted, introduced, linked, etc.) into the chain.
Add would be assumed at the end of the chain, although append would be more specific. Or you could say do D after C. With C already known to be part of the chain, it should not cause any confusion.
You would link D to C (or to the end of the chain). See the verb link in any dictionary.
The terms append and prepend are common in math and computer science for exactly this kind of thing.
D is appended to the chain.
D is prepended to the chain.
You can also say that the chain is prefixed or suffixed by D, but these sound awkward (to me) outside the context of natural language.