Here's the short answer: There is no chess table in the base game of The Sims 3. The game only offers chess via the Computer ("Play Game → Chess") or TV ("Play Game → Chess"). No physical board exists for Sims to sit at.
Place a chair at a computer desk. Your Sim sits, selects "Play Game → Chess," and gains Logic skill. This is the intended, functional method.
Place two chairs opposite each other at a dining table. Sims will not play chess autonomously here. You can only pose them for screenshots using testingcheatsenabled true and moveobjects on.
Large floor-piece sets exist as premium Store content or custom downloads.
For pure skill grinding: Computer → Play Game → Chess is the fastest. Alternatively, Telescope → Stargaze builds Logic and the hidden Astronomy skill.