Yes, when he uses his cypher try to spike him the best you can, also when he is recovering try to predict when he is going to use the chypher and spike him accordingly. By watching when he is going to use it, you can make sure that he doesn't just go as high as he can really quick so you can't get him. So by going early, you can kill him easy with the f air part that spikes, or the f air part that raeps them off the side of the stage.
Snake's cypher doesn't sweetspot the edge, so if he uses it and flies up in front of you grab his cypher and he will die. If you grab him while he's flying and not the cypher don't press anything, just let the grab release and he will just fall not being able to use his up b again. Proceed with the lulz.
I heard somewhere that if he was release grabbed he couldn't c4 recover. I could be wrong though.
K now enough with the cypher, the best approach option in my opinion is the rar'ing of double bairs. If he throws a grenade, try to catch it and throw it back. If he knows how to grenade strip screw it, don't do that again. The main focus of playing against Snake is to try to keep him in the air. Snake is juggled easily so use up tilt over and over until he's out of range and then use u air. Followed by another u air. It works suprisingly well. Also any time you ever see Snake vulnerable ever. Unleash a Giant Punch. In my opinion abusing the super armor against him works better than with alot of other characters. If you can predict something coming at close range, punch right through it.
Uhh...more advice...I think your f tilt is either at his level or better, so using that to bully him torward the edge is a good idea. Also if you can get him sitting near the edge, grab him, walk off and throw him against the stage. Anywhere off the edge is where DK shines. Another tip, use down b on his mines, they will asplode, maybe even catch him in it?