you'll need a flip-flop hub. if it's a nutted axle hub, the axle can be the same length and diameter as the one you have. if the axle on the new hub is a hollow axle designed for use with a quick release, the axle needs to be the same diameter, but will need to be shorter, depending on the inside width of your dropouts.

then the current hub will need to be removed from the rear wheel and the new hub installed, which may require a new set of spokes depending on the flange height of your new hub flanges.

i am skeptical as to whether your problems involving chain slack will be resolved with this modification though, spontaneous chain slack is usually caused by loose axle nuts and/or long term chain wear. but without looking at it, it's hard to tell for sure.