While an SSD will speed things up enormously, it doesn't tackle the immediate probable problem of the clogged cooling fan. Due to the basic design of the mainboard in the CQ56, if the fan is not doing its thing properly, the CPU and graphics chip are mutually heating each other to the point where failure can happen. An SSD reduces overall power consumption and thus heat generated by the entire system, but doesn't cause the two ICs that give the heating problem to run cooler, and its them wots focusing the problem.
Despite advances, SSDs are still limited in capacity compared with traditional hard drives, and, depending on what is on the machine, might not be big enough. The best first action must be to clean the accumulated muck out of the fan, then decide whether its worth money to upgrade to a SSD and the extra storage and whatever it will take to move from W7 should that wind up being needed. A CQ56, with the best will in the world, was never intended to be a top performer, rather a workhorse.