This interview with Christopher Bamford is a good introduction to some of the core (and, I believe, most interesting and important) ideas of Rudolf Steiner. Bamford is not a fluent speaker, but he is (I think) pleasant and engaging; once he gets past the first six minutes (which are mostly answering questions about about the sales of Steiner Press publications).
After a very good concise definition of the essence of Steiner's ideas; Bamford moves-onto a discussion of the 'so-called dead' as Steiner calls people who have biologically died.
I have found these ideas of Steiner's to be helpful in triggering my own thoughts on the subject - not by my agreeing with what Steiner believes happens after death (I regard him as mostly-wrong about this!) - but mainly by Steiner taking the matter seriously, and in the right spirit.
The right spirit is to recognize that there is one world that includes both the living and the 'so-called dead' - and also, I would add, pre-mortal spirits who have never been incarnated.
I think of this, however, in an individual way. I don't believe I am directly concerned with all the dead, or even most of them - but with certain specific people.
One point Bamford mentions is that the relation between living and SC-dead is potentially (and ideally) two-way and reciprocal.
Some of the dead remain concerned with, and interested by, the doings of some of the living.
And also we personally each have an interest in some of the dead - including some duties to them, because there are ways that some particular dead person needs you or me specifically - he or she can benefit from our actions.
Having established the general principles; how might I get to know who exactly needs my attention?
As usual in personal relationships, this does not work by formula. But I think sometimes I have had a strong and sustained inclination/ motivation to (in some way) pay attention to some particular person - a prime example is my fascination with William Arkle (which I have described here).
So - I believe that we can know this much without much difficulty. But if you were to ask me to tell-you exactly what we ought to be doing for a specific dead person; then I think we get into the usual difficulty with human relationships that any summary is grossly inadequate and can be dangerously misleading.
For example, if someone tried to discover what exactly was his duty to a mother, brother, wife or best friend... Then how could this (even in principle) be answered satisfactorily in any kind of explicit, comprehensive fashion? It could not.
And if you (unwisely) did summarize your duty in some sentence or list of bullet-points - then this reductionism of the personal would be far more likely to mislead and misdirect than to help.
The correct and simple answer to 'what is my duty?'; is something like loving attention.
Our duty to the specific-dead is loving attention... and whatever flows from that.