Have you checked to see if your record is warped? If not then there might just be micro-dust caught in the grooves that's causing a skip, something easily treatable, but barely noticeable until you listen to it.
As for cleaning needles there's a few ways out there, I'll go in order of price:
Mr. Clean Magic Eraser Cleaning Pads
Some people rely on these for all sorts of vinyl care (I do not though, but there are quite a few online who believe in these things). Just drop the needle onto a pad and immediately lift it up with the tone arm lever. you don't brush the needle with these things, the pad should pick up whatever gunk or dust is on the needle, if any.
Mobile Fidelity #9 Stylus Cleaning Fluid
I've heard good stuff about this one, but haven't tried it myself. Probably gets the job done and if its from a company like MFSL, then you're getting good products because they generally know what they're talking about.
ZERODUST ONZOW STYLUS TIP CLEANER
Now this is the real deal. I've used it twice now and really like this stuff. It's a proprietary gunk that like the magic pad you drop the needle into and immediately lift up. Then you stir it around the the dust or gunk falls to the bottom. Works great and lasts for YEARS.
As for cleaning records themselves...
Mr. Clean Magic Eraser Cleaning Pads
I've actually tried this on a junky record with no attachment to and the results weren't stellar, but it sorta did ok? I dunno, proceed with caution, try it on a dime record and see if you like the results.
Mobile Fidelity: Record Cleaning Brush
This is my favorite dry cleaner. Picks up any surface dust, paper or dirt. While it's not a thorough cleaning it gets the job done for newer records or records that have been previously water cleaned.
In The Groove Record Cleaner
Have never tried them myself, but some people swear by the results. I'm paranoid about any sticky texture being applied to my vinyl though.
Hunt: EDA Mark 6 Carbon Fiber Record Brush
Something else people often swear by is a Carbon Fiber Brush. Not my cup of tea, but I think there's a huge amount of record collectors that love these things.
SPIN-CLEAN
This is my favorite deep clean, wet vinyl care system. I've had one for almost a year and have gone through my collection extensively cleaning and replacing the inner sleeves on everything with non-paper sleeves. Easily recommended if you imagine yourself really getting deep. Otherwise it's a big price to chew if you're just starting. The solution included, if used properly will clean 50 records a session, so you'd need some investment before buying into this.
Nitty Gritty 2.0
This is my dream. Couple this with the Spin Clean, you'd get the one two combo of a water clean plus a vaccuum to really get those records back to their glory. Nothing will fix scratches or dents, but this is damn well near perfect.