The sound drivers are not included in the dCore kernels but can be imported through the dCore extension alsa-modules-<kernel>-tinycore. The "README"-page about ALSA provides a complete walk-through.

• The TinyCore-wiki contains four articles about setting up sound, two (one and another one) about ALSA plus one about OSS and one about PulseAudio.
• Sound management is a complicated issue, indeed. This can be seen from the fact that just about every distribution gives it ample consideration in their wiki. The Archlinux-wiki, for instance, is an excellent source of information.
• Just a short note to new-comers about trouble-shooting from a former new-comer: In case everything is set up well but nothing works it is worth checking if the channels do not happen to be muted. Alsamixer, for instance, will show the volume bars turned up but mark a muted channel by a letter M below it. This is easy to miss (trust me, or, if not, cf. screen shot).