This is a tricky one, and I may be painting myself into a corner in which case please feel free to tell me so in an answer but please also provide advice on how to get out whilst disturbing as little paint as possible!

I'm 5 sessions into a new game I've never played before and I've started fleshing out my character's background. I love detail so I'm attempting to tie it all into her class, her stats, her race, everything but I'm new to the game and I don't want to learn too much about the in game universe that will ruin surprises / locations / plot points / trivia that will likely be revealed in game (or in games to come).

For me, learning about the in game universe organically (as opposed to reading about it outside of the game) is one of the huge aspects of my enjoyments, and I don't want to spoil that.

How can I tie my character's background into a game I know little about?

I've deliberately left out the game details as I think advice could be system agnostic, and I've left out specifics of the detail I've allowed myself to learn out-of-game so far as I feel answers that give the minimum possible would be of more help to more people (who can then make their own concessions).

However, if it aids your answer: I am playing DnD5e, I've allowed myself a map of Faerün (although I am not looking up any of the places named on the map) I'm trying to tie in the loss of my orc father in a war (what war? where? I've selected a place but can I really make up my own war?) and I'm claiming I learn Dwarven and Elvish on route to find him (as a result of a chance encounter with some Elves at Location X, but what if there is no way Elves could make their way to Location X, in fact Location X's one defining characteristic is that it has never had Elves in it ever!).

It's time to talk to your DM. You clearly have ideas about the general shape of your backstory, but you need to know some things that "everyone in the setting knows" and you need names, places and dates that don't conflict with the setting's plots, or the DM's own ideas for their campaign.

It may well be that the DM will use your backstory to help develop the place(s) your character has lived. For example, if someone in your home village encouraged you to take on a particular career, that's part of a NPC designed. Dungeon World makes a point of delving into such matters, but the idea is applicable to just about any RPG.

Working with you on this is worthwhile for the DM, because it means they can reveal things in the course of play that "your character has known for a long time," but become significant in the light of campaign events. They can also have events that tie into your backstory, but they should not have too many of those, nor drive the plot by them, because it will make you feel like their puppet, rather than a contributor in your own right.

Work with your GM

The key thing is to work with your GM to determine how you should be linked to certain things and how much you and your character should know. This can be an ongoing conversation and your background can change and grow in depth as the story develops. If you like you can think of those additions to your background story as akin to a literary flashback.

Your character likely knows a bit of setting lore and you should too.

You mentioned you want to avoid spoilers, and to a certain extent that is a good thing, but remember that in most scenarios your character will know a fair bit of the lore of their world and will likely know their own personal history.

There are of course exceptions to that. A pseudo-medieval peasant may actually know very little of what is going on 20 miles away and what they do know may be exaggerated. But in most games the PCs are assumed to be more competent than that and are generally in some way special. Also, even an illiterate psuedo-medieval peasant that has never traveled will know his own hometown.

In short, as long as you aren't delving into spoilers about your specific campaign, a certain amount of background knowledge is likely to be expected and helpful. Again, you can work with your GM on the details.

Have your character be a recent arrival.

Again work with your GM, but more specifically, one of the easiest ways to flesh out a background without worrying about spoilers is to have your character be a recent arrival to the region where most of the action will take place. Then you can detail all kinds of adventures for your character before the start of the game without worrying about interfering with the GM's plans because they all happened over there. The GM might then decide to have some of those loose ends follow you, but that will be the GMs call. The GM is equally free to leave them all as the way your character discusses his own history.

Appropriate Background Lore can help avoid spoilers

In most tabletop RPGs, your character is an experienced native of the game universe. This gives the DM lots of power in storytelling. If magic is rare in your universe and you encounter an adversary bedecked in glowing rings, you immediately know you're up against a real threat. But your character, and thus you, should know the general religious, cultural, and political background as well.

Consider James Bond type films, where avoiding spoilers to the villains' nefarious plots is definitely part of enjoying the movie. We still know an incredible amount of general background that lets the filmmaker get on with the story. The Prime Minister and the Pope are important people; North Korea is a dangerously unstable totalitarian regime; SPECTRE is the secret villain's club. Now transport that to a fantasy realm. If your DM has to pause the action to point out to you that the Steward of Gandar is the second most powerful person in the realm, it puts a spotlight on that NPC. "Oh, that guy must be important later!". Whereas if you and your character simply know the general background to the world, we don't get that effect.

Avoiding lore spoilers is a noble and desirable goal. Fortunately, D&D 5e (and many other game systems) facilitates this by segregating player resources from dungeon master resources. The racial descriptions in the PHB strike a very nice balance in information disclosure - enough to concoct an appropriate background, but not so much that the world becomes demystified (as might happen if you started reading through the Forgotten Realms wiki or a campaign setting book). As a baseline, of course you should read your own character's race page. In addition to this, if your character has had contact with other races it would be entirely appropriate for you to read the entries for those races as well. The information is vague enough that an outside observer of said race would reasonably know it.

For places and events, player resources are not going to be as useful. Some racial descriptions provide a few homeland names, but historical details are very sparse (likely by design, to avoid spoilers and overwhelming). This is where you need to speak with your DM, the ultimate keeper of the lore. Even if you are able to develop a background that meshes into existing lore all on your own, it's important to consult the DM to (a) make sure you have the lore right; prewritten lore is always subject to modification by the DM (b) make sure your background can be worked into the game narrative. The fastest way to do this is to tell throw the ball entirely in the DM's court: give them the rough template of your character and allow them to fill in the gaps. This has the great benefit of allowing the DM to more easily incorporate your background into the game narrative. Parents killed by a liche? Let the DM come up with the details of who that liche was; perhaps they can use a villain that they plan on introducing later. Born in a swamp? Let the DM pick the swamp; perhaps there's already a quest planned that will bring your party to a swampy area in the future. This all depends on the amount of work your DM wants to do, of course, and they might decide to kick the ball right back into your court. If that's the case, there's not much else you can do but make your background more generic. For example, instead of a famous war, your father dies in a routine skirmish.

Working with the DM is the obvious solution.

The alternative is to make your character from somewhere very far away from the story, where you can make up a location and it won't interfere with the canon lore. Maybe you met a pair of elves as part of a trading caravan passing through a little village in a corner of the biggest empire in Faerun. Maybe you showed up in canon-lore land just days before the game started and know absolutely nothing about the local people or its history. This would actually give the DM some leeway in introducing a personal quest for you, like finding the caravan elves at a big festival in town and hearing how the caravan got robbed, or your homeland was hit by a disaster and there's a lot of refugees pouring in.

This is a fantasy world. You can go a bit crazy. If your DM will let you, have your character arrive from a different world. They could native to Eberron, chasing their dad and following his trail into an old cave. They go in, find some crazy artifact, and boom, now they're on Faerun and they realize that their dad never came home because he was stuck somewhere in Faerun.

Depending on the DM, they can help shape your backstory to fit the lore, or even shape the lore to fit your backstory! The DM can absolutely say "Elves don't usually show up in location X, but it was one of those rare occasions where they were forced to go there to get supplies."