In the original House of Cards series, Corder leaks a tape and documents to Makepeace. I don't see the reason for doing so. Killing Urquhart sort of made sense, but why leak the negative information to his rivals?
I don't have official source for it, but for me the following explanation makes sense:
In the last dialoge of the series, Corder speaks to Makepeace:
Everything's under control.
You'll be in charge now.
Anything you need, we're right behind you.
So, Corder seems to feel that Urquhart will no longer be the leading horse. If Urquhart gets accused for murder, he might be at least a minor suspect (in terms of helping Urquhart). The leaking of the tape to Makepeace together with the killing of Urquhart is the only way to preserve Urquhart's reputation, to not beging an investigation, and to show the future leader Makepeace his loyalty (hence saving his job and power).
Since Makepeace was talking about advantages for Urquhart or a family member, it is possible that Makepeace would be investating everything even after Urquhart has died. Makepeace knows the truth now, and finds out that his own family (At least Corder) has betrayed Urquhart. So, this action of Corder was also a favor for Elisabeth (They do not share much screentime together, but we find out that they are pretty close).