Yesterday, I started watching a TV show called ''The House of Cards''.

It's a relatively famous series, there's no "The" in its title.

The main charachter is played by Kevin Spacey, whose acting I greatly enjoyed in "American Beauty''.

In the first episode, we are introduced to ''Tthe 'House of Cards'' - the Executive MansionWhite House.

"House of Cards" is partly a pun on the name of the White "House", so it's sort of weird to rephrase that. While there is a part of the White House complex that's referred to as the Executive Residence, most English speakers would just refer to the White House in general as its a famous location.

The narrator, a congressman who ''latched onto him (the President) early on and made himself vital'', had been granted a promotion to Secretary of State.

However, despite the chief's(President's/head of state's) promise, and much to the surprise of everyone in the White House, another candidate is selected.

While the President is the head of state for the US, he'd normally be referred to as "the President" or maybe "the US President" if you needed to distinguish between them and some other country's President.

The main charachter is livid; he resolves to avenge himself slowly and accuratecarefully, adroitly employing all of his power and connections.

"accurately" is not a natural phrase to use here.

So far, I have really enjoyed both the seriousness and the humor of the show.

It is directed by David Fincher, who is probably one of my favorite directors after Quentin Tarantino.

Your original sentence would work when speaking English but when writing you need to put "who is" or "he is".

The narrator, a congressman who ''latched onto him (the President) early on and made himself vital'', hads been granted promotion to Secretary of State.

However, despite the cChief's of sState's promise, and much to the surprise of everyone in the White House, another candidate is selected.

Chief of State is his role so should be capitalised. Good use of the apostrophe as it is "his" promise, it just needs to go after "State" as that is his title.

So far, I have really enjoyed both the seriousness and the humor of the show.

The original sentence works well, I have added in "have" so it reads nicer.