I've been watching some videos on the Metal Slug series, and it looks fun.

Is there any way to try these or something similar on the PC?

Move to the lounge.

Neo-Geo emulation has been pretty solid for just about forever since its architecture isn't especially complicated and pretty much all the ICs were of-the-shelf (save, perhaps, the graphics hardware). The legality of downloading the necessary ROMs is somewhat grey if you own an actual copy of the software (and the hardware, in the case of the Neo-Geo firmware), but quite solidly black if not. Not that this stops anyone. We don't really condone linking/recommending Roms/Warez sites on these boards, but any search engine will turn up results.

On the perfectly-legal front, Alien Hominid is very much similar in gameplay to metal slug, and started its life as a Flash game (on NewGrounds.com, IIRC) before its various ports to the gamecube and Xbox Live Arcade -- it's probably still available to play entirely free.

The first 4 Metal Slugs (Metal Slug, Metal Slug 2, Metal Slug X, Metal Slug 3) were apparently released for Windows, according to their respective Wikis but I'm having a hard time finding a place to buy even the first one.

Metal Slug Anthology is available for Wii, PS2 and PSP. That contains Metal Slug 1 ~ 6, including X. Probably the cheapest way to go.

There's also gametap. Clicky.

If you're inexpensive like me, there's also newgrounds ripoffs. Clicky