I'm pretty new to D&D, and right now I'm trying to make a character starting at a higher level for a campaign. The DM is allowing us to start with some magical items of our choice.

Using the Wayfinder's Guide to Eberron, I decided to make a Warforged, and the Warforged race has a few exclusive magical items, one of which is the armblade. In the description of the armblade, it says:

An armblade isn’t inherently considered to be a magic weapon for purposes of overcoming damage resistance. However, any sort of magical melee weapon could be created as an armblade, so you could acquire a vicious armblade or a vorpal armblade.

However, as the armblade already requires attunement, if I make a magical weapon that already requires attunement as an armblade, does the resulting weapon require double attunement or something?

No, attunement is attunement

An item can either require attunement or not. The armblade requiring attunement simply means the weapon requires attunement (as in attaching it to yourself) even if the normal variant of the magical weapon doesn't.

Specifically for the "number of attunements", the restriction is on the number of magical items not the number of attunement, as per the DMG (p. 138):

An item can be attuned to only one creature at a time and a creature can be attuned to no more than three magic items at a time.

No, you only attune once

..., any sort of magical melee weapon could be created as an armblade, so you could acquire a vicious armblade or a vorpal armblade.

The above phrase simply refers to the fact that an armblade can be enchanted in the same way as any other magical weapon.

That is, magical weapons in the Dungeon's Masters guide (and other sources) state the kinds of weapons that can hold a given type of enchantment.

For example, "Vicious Weapon" says "Weapon(any)". Meaning that potentially any kind of weapon could be found that has this type of magic: A vicious longsword, a vicious warhammer and so on.

So, what this is saying is that you could get a vicious armblade in the same way that you could get a vicious longsword. But even if the armblade always requires attunement anyway, this has no additional effect, since you are only limited by the number of items you can attune (not 'enchantments').