We do this with a simple anchor trolley, which is nothing more than some line that makes a loop on the side of the kayak that will take the anchor line to the bow or stern. Your trolley has a clip (usually a thumb snap) on it to run the line through then work the pulley (just like you mother might have worked the cloths line pulley) so the line is fore or aft on the kayak. You then have to tie off the line on some kind of cleat.
Clam Cleat ZigZag Cleat
Most kayak fisherman are using a either a 1.5 lb or 3lb anchor depending on the application and amount of current, the holding power of any anchor can increased by adding a foot of chain, this allows the anchor to tilt in a better position on the bottom and allows it to grab faster with less line out. Usually to anchor you need to get the line out at a 45% angle to grab properly. It's also possible to use a drift chute instead of anchor.
1.5 lb folding anchor Drift Chute
Sometimes keeping your line organized on the kayak can be a challenge, one way to go is to wrap your line around the cleat you will use to tie-off can work, you can wrap about 15-- 25' of 3/16 line on a 5 inch cleat. Another option is wrapping the line around something like a rope reel that easily wind up the line. The last thing you want to do is be untangling your line when you are in an area that you want to fish. You anchor system can also be used for drift socks or even a stake out poll if you are fishing the flats. Personally, I keep my system very simple with the pulley only taking the anchor line to the stern of the kayak but some people prefer the option of going to the bow also. If you anchor off the stern and then decide you would like be facing and fishing the opposite direction, you would just move the anchor clip to the bow and you will now be turned 180 without re-anchoring - this can be useful while kayak fishing.
When it comes to rigging the system you can use deck loops to act as guides for the line. A 3/16" braided line will work well. Another option is to use a piece of bungee at one or both ends of the system to provide some shock absorption while your anchored. Another option is a pulley (block) to help provide smoother operation of the anchor system. Sometimes you might want to undo the anchor and leave the line and maybe return to that spot, a small buoy float works for this, just run the anchor line though the float and as long as you have a clip on one end of the line the buoy will work its way up the line but not come off. An anchor trolley can work that same way for a drift sock or a stake out poll.
Anchor deployed (Note: line passes through trolley clip)
Anchor deployed from bow