You should read up on how an anplifier like this works so you understand what each of the components and values do.
Your new schematic is correct although why you want to run it from 3.6V when the DTMF IC needs 5V is unclear, they will both work fine from 5V so link their supply pins together. The detached 10uF capacitor is not needed, the one across the supply only has to span the + and ground pins.
The two 100K resistors are there to set the voltage on the output pin. Because they are the same value, the voltage at their junction with the IC will be at half the supply voltage and in this configuration it will make the output pin at half voltage too. This is ideal because it allows the output voltage to swing equally in both directions, toward ground and toward supply. Looking at it a different way, the output pin should be at 2.5V so it can theoretically go 2.5 above that (5V) or 2.5 below it (0V) giving you as much as 5V of signal but if the pin was at 1V, it could only go down as low as ground (0V) so the signal would have it's bottom clipped off.
It's an inverting configuration with feedback (= gain setting) controlled by the amount of signal fed back from the output to the inverting input. If you directly linked the output pin to the inverting input you would get a gain of x1 (same out as in) but if you reduce the amount of feedback the gain becomes higher. The resistors connected to the inverting inut are there to set the amount of feedback and therefore the gain. The capacitors have two jobs, the first is to block DC so the steady voltage on the inverting input comes only from the output pin, this stabilizes the operating conditions inside the amplifier IC. Their other job is to control the frequency response. Remember that capacitors have a property called reactance (symbol Xc) which is similar to resistance but it changes according to frequency, Xc = 1/(2 * pi * f * C), so as frequency increases the reactance falls. The capacitor between the output and inverting input (across your variable resistor) makes the gain lower as frequency increases and the one in the inverting input to ground makes the gain increase with frequency.
I leave it as an exercise for you to do: using the formula for Xc, and the formulas for resistors in series and parallel and assuming you can treat Xc as a resistor (almost true) work out what the feedback and gain are at different frequencies. The range you are interested in is from about 100Hz to 10KHz so pick some frequencies between them and draw a graph of the gain. It think that will explain why I consider the capacitor values to be wrong. Don't forget frequency units are Hz, capacitor values are in F and resistors are in Ohms.
Brian.