You don't need to do anything.
Halo 1-3 is the only important story, the extra content is just fluff and quite honestly a waste of time if you're simply looking to make your gameplay better.
The games are designed so that, if you completed the last game, you're all good to go.
If you want background on the chief, here you go
https://en.wikipedia.org/wiki/Halo:_The_Fall_of_Reach#Plot
If you want to expand further:
http://halo.wikia.com/wiki/John-117#Biography
That's it, 2 wikipedia pages to get you up to scratch, The contents of any of the books do not add to character or influence the plot at all.
Play through Halo 1-3, Get to the end of Halo 3 and hang tight.
New characters spring up from out of nowhere in Halo 4. They are practically useless to the story but because they are defined in earlier works (Books, TV shows etc) , they aren't really given an introduction. Not that it matters.
To summarize (No spoilers I promise).
UNSC Infinity: It's just a ship, It could be replaced with any other ship and still work into the plot, treat it as the Pillar of Autumn without the crash landing.
Captain thomas lasky: The guy who is the co-captain of the ship, Kind of sympathetic towards the chief after he got saved in his camp in the live action thing (FOD), definitely not worth watching.
Captain del rio: The captain of the ship, Severely unimportant, don't even bother researching him.
Sarah palmer: A spartan like the masterchief, No reason to get attached to the character, Play spartan ops and you'll want to drive a dagger through her head.
Speaking of spartan ops, Once you have played through Halo 4's campaign, You'll have these to play.
Don't think too much of them, it's a shitshow of going to X location to fight Y enemies to going to Z location to press a button and leave again, it's not fun, it's a grind. To prove my point, 343i aren't making another season at all.
Best chapters are 6 and 10, if you do decide to play through any, these are the ones. General consensus is that some elite captures Halsey at the end.
Onto Halo 5.
Go through the entire campaign, and that's it.
No more books to read, no more anything. You will have caught up.