You need to play all the main series games in release order. If all you have is a 360, then your options are quite limited.
MGS1 - Emulate on PC, only the PS1 version; Twin Snakes, while having nicer graphics and mechanics, suffers from the game design not being adapted to fit the updated mechanics, which ends up breaking a lot of things (like Ocelot's boss fight).
MGS2 - On the HD collection for 360
MGS3 - On the HD collection for 360
MGS4 - This game is only on PS3, so you don't seem to have any options. You can watch it on YouTube, but despite how extremely long the cutscenes are, it has some of the best gameplay in the series (for at maybe 2 or 3 of the 5 acts).
MGSPW - On the HD collection for 360; people say skip this, but I say DON'T. MGSV is a direct sequel to the events of the Peace Walker so it's absolutely necessary that you play it.
MGSV - Both Ground Zeroes and Phantom Pain are on 360.