So I'm level 2 now and as a fighter, I'm close to getting my martial archetype. I am without a doubt picking Eldritch Knight but want to know how he could get at least one spell that is 9th level. How would one go about that?
It is possible at 20th level if you multiclass.
The problem is, in order to get a fighter's martial archetype, you would need at least 3 levels in the fighter class.
This means you would need 3 levels in fighter + 17 levels in another class. Wizard would be the best choice, because it is Intelligence-based, like your Eldritch Knight.
In total, that's 3 + 17 = 20 levels! And getting there (from your current level of 2) may take a very long time.
Note: It is true that if you multiclass in two spellcasting classes, D&D 5E lets you combine parts of both class's spellcasting abilities. However, even if you get access to a 9th level spell slot, multiclassing won't necessarily give you access to a 9th level spell.
If you have more than one spellcasting class, this table might give you Spell Slots of a level that is higher than the Spells you know or can prepare. You can use those slots, but only to cast your lower-level Spells. If a lower-level spell that you cast, like Burning Hands, has an enhanced effect when cast using a higher-level slot, you can use the enhanced effect, even though you don’t have any Spells of that higher level.
For example: Multiclassing Fighter (Eldritch Knight) 3 / Wizard 16 would result in the equivalent of a 17th level spellcaster. They would have a 9th level spell slot, but they don't have any 9th level spells to use until they gain another level and use it to reach Wizard 17.
Eldritch Knights only have spell slots up to a maximum of 4th level at Fighter level 19-20. They (along with Arcane Tricksters) are "one-third casters", meaning they gain higher-level spell slots at one-third the rate of a "full caster" such as a wizard.
The only way an Eldritch Knight would gain 9th-level spell slots is by multiclassing into a full caster class (wizard, druid, cleric, bard, sorcerer).
Single-class full casters gain 9th-level slots at class level 17.
Per the multiclassing rules, a 3rd-level Eldritch Knight would contribute one level towards the spell progression of a multiclass spellcaster, so multiclassing into a full caster (and taking only levels in full caster classes) would let you gain one 9th-level spell slot once you hit level 16 in that class. However, since those rules specify that "You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class", you would not actually learn a 9th-level spell until you hit level 17 in the full caster class.
Spell Scrolls can partially replicate high level spell-casting for an EK
An Eldritch Knight can cast spells from scrolls which are higher level than he can cast normally, according to Jeremy Crawford (official D&D rules guru) (https://twitter.com/JeremyECrawford/status/537374983576498176).
Please be aware that there is a very high chance of spell failure when trying to over-cast from spell scrolls, since you need to pass an ability check with a DC of 19 (10 + spell level), using your spell-casting modifier. A Headband of Intellect and some method of getting advantage on Intelligence checks (such as the Enhance Ability spell) are recommended.
9th level spell scrolls are also of legendary magic item rarity, which means they will be very hard to find, depending on your DM and style of game.