The not-as-good news: The research I have done shows that there is no out-of-the-box d3 library that directly accomplishes this without some customization.
The good news: There have been some other people who have looked into this and have found some great starting points! I realize that this is not a complete solution to the entire task at hand, but it seems from your question that a large portion of your difficulty so far has been simply figuring out where to start (e.g. "Tutorials about d3.js only cover standard charts like bar charts."). In the absence of anything better, I will at least respond to that portion here.
First, in the response to this related stackoverflow post from a few years back, inanutshellus provides some great d3 tools that are available and could be of use here. With some light customization/extension, they should be able to get you where you're going relatively quickly. For posterity, inanutshellus's answer is reproduced here:
There are some options, but I believe each would require a bit of
work. It would help if there were one single standard for representing
a family tree in JSON. I've recently noticed that geni.com has a quite
in-depth API for this. Perhaps coding against their API would be a
good idea for reusability...
-- Pedigree tree --
The Pedigree Tree
might be sufficient for your needs. You'd make in-law's linkable,
where if you clicked on their name the graph would redraw so you could
see their lineage.
-- Bracket Layout Tree --
Similar to the Pedigree Tree, but bidirectional, this Bracket Layout Tree
lets you handle a "here are my parents, grandparents, children,
grandchildren" type view. Like the Pedigree Tree, you'd make
individuals linkable to re-center the bracket on that node.
-- Force-Based Layout --
There are some interesting force-based layouts that seem promising.
Take a look at this example of a
force-based layout with smart labels. An adjustment to the
algorithm for how the "force" is determined could make this into a
very lovely tree, with older generations above or below newer ones.
-- Cluster Dendogram (why it fails) --
The d3.js layouts I've seen that would lend themselves best to family
trees assume a single node is the parent, whereas you need to
represent the parent as the combination of (visually a "T" between)
two nodes: one node that is a member of your tree, and one floating
node that represents the in-law. Adjusting a cluster dendogram to do
this should be feasible but not without significant modification.
If you--or anyone else for that matter--tackle this, let me know. I'd
like to see (and benefit from) the work and may be able to contribute
to it if feasible.
In terms of concrete implementation, mj8591 asked this question regarding a similar family tree with a different problem. However, luckily for you that question includes a fiddle (all the js code) that has most or all the components that you need, and the response from mdml includes another fiddle that adds some more granular "clickability" to each node.
Again, it's nothing automagic but hopefully these resources are enough to get you a great start!