I've had luck opening various remotes by running my thumb fingernail around the seam and gently prying more and more until it pops apart. But first do open the battery compartment and check it carefully for screws. Feel any labels for a hole under them, indicating a possible hiding place for a screw.
Your typical remote uses plastic clips molded into the body to hold the top and bottom halves together. Sometimes, you might need to push in either the top or bottom edge along the seam to help pop the clips apart.
By the way, once you get it open, you might find out that just cleaning doesn't really help the problem (sometimes it does). Many remotes have rubber buttons that are impregnated with carbon on the bottom to act as a switch as it pushes against the circuit board pads. Sometimes the carbon just comes off over time. I've seen a remote repair kit for sale which contains something that you paint on to the bottom of the buttons. I've used conductive ink to good effect as well. Once I tried putting little pieces of aluminum foil under the buttons, but they didn't tend to stick on very well.