There's a few non-raid upgrades you can make to your gear still. Balefuls, Honor gear, whatever. Honestly though, that's minor.

At your ilvl, you shouldn't be having too hard a time getting into Normal mode groups. While some do ask for the achievements or the ring, it's not terribly common. As far as your gear is concerned, you have not yet progressed past even lower HFC. You have no tier pieces, unimpressive trinkets, and your off-hand weapon and bracers are very low ilvl. Many of those things can be fixed in lower HFC.

I think something you may be missing about raiding is that "kill each boss once or twice then you're done" is very rarely how raids operate - especially for PUGs. It's about repeating the early stuff for a while to get the gear to be ready for later bosses. Assuming a similar gear entry point, an entire raid group of people with your gear level and experience (max 2 kills each boss) is unlikely to make it all the way to Mannoroth/Archimonde without resetting and clearing a couple of times for more gear.

Anyway, if you're really not into that and you just want the Archimonde kill (and can't get with a guild as you'd said elsewhere), here are your options:

1. Sneak into a guild run that's short a few. Refresh the search results a lot. The trick is to find a group that has no bosses killed at all, but already a bunch of people, and has only been listed for a minute or two. I've gotten several full clear groups this way on alts at around your gear level.
2. Check OpenRaid.us (or OpenRaid.eu). It's a website for organizing cross-server raids. I've seen and been in a few groups that were definitely about helping people get Archimonde clears. You need to adhere to a schedule, but you sign up to raids, so it's according to what you're available for.
3. Buy an Archimonde carry. You'll have the achievement, though if anyone looks at your gear, they'll know you were a carry since most sales are for Heroic Archimonde.
4. Create an Archimonde "progress" run. Make sure it's labelled as a group for people learning the fight. You might get some help, and it may work. It may also fail miserably, so only try this if you're good at shrugging off inappropriately aggressive player behaviour.