Water flowing continuously in toilet

If your tank is constantly leaking water, but you've not found it anywhere, then it is nigh on certain that it's going into the toilet bowl. Tester101's tip of putting food colouring into the tank and seeing if it appears in the bowl will likely confirm this.

The issue is then to find exactly where the leak is. Either your flapper isn't closing properly, there's a leaky joint where the overflow pipe connects (just below the flapper), or the seal between the flapper apparatus and the tank isn't watertight.

I would hazard a guess at it being the seal between the flapper apparatus and the tank, probably due to either under-tightening or over-tightening when installing the new flush. Over-tightening is perhaps the more likely - it shouldn't be tightened too much or you will distort the rubber seal.

Given a recent question, you might want to consider replacing the whole toilet. There are good examples of low-water-use toilets out now that reliably flush with far less water (1.28 gallons per flush) than older toilets, and you know all the parts will fit.

Otherwise you are looking at replacing, in order of ease, the flush valve or flapper (usually a rubbery part - they get old and crack or stiffen), or the flush valve seat and overflow tube (which often comes with a new valve) and pretty much always requires taking the tank off the bowl (cistern/pan for the across the pond folks.) Since you have to take the tank off the bowl, you'll want replacement gaskets for the tank to bowl screws.

That is assuming that you don't have a very simple issue such as a mis-adjusted float causing water to go over the top of the overflow tube - where you'd need to adjust the float, or replace a faulty fill valve if it can't be adjusted.

Aside from reduced water use, the main thing that buying a whole new toilet gets you is freedom from the multiplicity of different makes and models of toilet with non-interchangable parts. You want to know the brand (at least) of your toilet when you go shopping, and having some measurements or taking the defective part with you can help to sort out which parts will actually fit your particular toilet.