Look for visible signs of pooling water around the drain field area of the tank. This indicates that the septic tank is getting too full of solid materials, causing the drain field pipes to clog and waste water to seep out onto the surface.
Walk in and around the perimeter of your house and see if you can detect any foul odors that shouldn't be there. A full tank can cause unpleasant smells to waft into your otherwise clean home, and outside they will be especially noticeable near a failing septic tank and its drain field.
Pay attention to drains that take a particularly long time to empty. If you find yourself using clog removers or toilet plungers on a frequent basis, your septic tank is not able to properly process the water passing into it.
Examine the grass growing around the drain field and septic tank. If it is growing at an enhanced rate, chances are the tank is spilling over and fertilizing the grass.