If I am using a water-pump filter. Is it necessary to combine this with a water purification tablet?
As discussed in this answer and at greater length here, the need to treat backcountry water before drinking it is largely a myth. Neither the filter nor the tablets are needed. You're better off focusing your efforts on avoiding the real reason that backpackers are likely to get sick, which is hand-to-mouth contamination. Practice good potty hygiene, and don't share pots with your hiking partners (whose gut flora your body hasn't got a tolerance for).
If you're traveling in the third world, or drinking water that you collected downstream from a poultry farm in Arkansas, that's a different matter. Filters work against protozoan cysts. Tablets kill bacteria and viruses. Another good option to consider is a steripen, which treats water using UV light. It kills cysts, bacteria, and viruses, is more compact than a filter, and is faster than tablets.
Katadyn, a water filter company, suggests doing so.
Katadyn sells tablets, so they have an economic interest in propagating myths about the need for water treatment.
However, this makes me question the safety of putting such chemicals into my drinking water. In general, is it best to just avoid it? Or are there any water purification tablets that are proven safe?
Iodine is a nutrient that your body needs. That's why they iodize salt. So the question is not really whether iodine is bad for you but whether a certain amount is good or bad for you. The lethal dose in humans is a couple of grams, and it kills you by oxidizing proteins. A typical concentration used for purifying drinking water is about 3 mg/liter, so by drinking a liter of water you're getting about a thousandth of the lethal dose. For comparison, a gram of iodized salt has about 0.02 mg of iodine.