When you say "cleaning", what kind of dirt do you want to remove?

It looks like someone tried to make it less wobbly by putting two boards across the back, but that doesn't help much.

To "tighten it up", you will have to take it apart and clean and re-glue the joints, with proper clamping. If any bits are structurally unsound, gluing in bracing blocks will help.

TSP is VERY hard on wood.

It bleaches some types, and turns others nearly black.

Paint thinner is safer for the wood, but needs good ventilation since it is flammable.

I do not want to remove or damage or alter the crackle/aged effect of the painted wood. So, would paint thinner remove the paint?

I mainly want to remove surface dirt of the front of the cabinet, and maybe do something to preserve the look of the crackling paint (ie: clean it and put on a coat of poly)

I want to make this cabinet look better, fresher but without altering it's aged look and affecting it's resale value.