Not well at all. DK is prone to painful, agonizing failure against characters with only decent juggle games, but against a character like Mewtwo with one of the best juggling toolsets in the game, it's even worse. Luckily Shadowball isn't too good of a projectile so it's hard to get camped, but in general you can't make many errors against higher level Mewtwo players. DK doesn't have the tools to properly edgeguard Mewtwo's kickass recovery, although I'm not sure many characters do anyway.
Play it like any floaty matchup; choose your combo opportunities carefully and try not to rely on cargo uthrow too much, because it's not a guaranteed combo starter like it is in other weight/fallspeed classes. Mewtwo's got great range too, but don't hesitate to throw out ftilts and dtilts if you need breathing room. Like other floaties, plain old downthrow can lead to great followups at mid-high percents. Uair is a more of a kill move than a juggling tool against Mewtwo since he's so light and floaty. Treat it like Fox's uair, and you can surprise yourself with how early it'll kill.
Giant Punch is a great tool to use in any light floaty matchup because it'll kill so early, and it's decently fast and ends pretty quickly. If your opponent isn't too aggressive and doesn't seem like he wants to approach, don't hesitate to circle camp the stage a bit and charge your punch with quick single swings. This is actually really easy to do on bigger stages like Dream Land, Dracula's Castle, Distant Planet, Skyloft, and Skyworld, although I wouldn't particularly recommend taking a Mewtwo to any of those places. Having a Giant Punch in your pocket gives you a lot of options for early KOs that are so important against floaties, who are really tough to chain attacks on at higher percents.
Also, on some of the smaller stages, backthrow is a viable KO move because of Mewtwo's really low weight. Because DK spends most of his time using his cargo throws, people will rarely DI the backthrow correctly if they don't see it coming.