Split a single branch circuit into two runs?
Yes, that's fine, you can "tee" circuits.   I just did one that looks like a tree... starting with a 3-way split right in the service panel.  And two branches themselves have splits downstream. 

Now let's talk about splicing.  USA receptacles provide two screws or backstabs per side.  That has several uses, but the most common use is an easy way to do a 3-way splice: one for the power source, one for the downstream feed to the next outlet, and of course the receptacle itself.  Here's the thing: people get hung up on this, and forget this is only one of several ways to do a splice.  You can use any listed method. 

So with both screws full, how do you add a third wire?


you cannot use both screws and backstabs.   You must use one system or the other only, and don't use backstabs anyway becuase they fail a lot. 
You can use a short pigtail to come off one screw, and then any of a variety of splices to join the pigtail and two of the wires.  One remains on the other screw (or not; you can just join all the wires at the splice, I do that when the receptacle is in a really awkward location.)
you can use special receptacles called screw-and-clamp which can accommodate four wires (plus the receptacle itself obviously).  You have to hunt these down, but they cost about $3 - not unbearable. 


What kind of splices would you use? Any listed splice.  Some examples are wire nuts -- Alumiconns (they work fine on all copper wire) --  Euro type terminal blocks -- push-in (unreliable) -- snap-down splice connectors.  I use wire nuts myself.