Check and make sure your alternator belt is tight and not slipping, but it sounds like bad alternator. Starter/Battery/Alternator are like a family. When one part has problems it can put a strain on the others. For example, a bad starter might make you crank a lot, which drains battery, then the alternator is overworked constantly trying to charge up a weak battery. So it's not uncommon for multiple parts to fail around the same time.

When you turn the key to on, 12+V is applied to battery light on the dash, which makes it light up. When you start the engine, a properly functioning alternator ALSO applies 12+V to the dash light bulb...it's like a head-on collision of power along the same wire. If the alternator is working right, the two forces cancel each other out and you end up with roughly 0V at the light bulb, and it goes out. If alternator is weak and puts out say 8V, it's not enough to cancel out the 12V, so you end up with 4V at the light bulb which equals dim/flickering bulb.