How do I reset a capacitor once I have a signal rise?
What you are doing is called a "Time over Threshold" measurement that is very common in particle detection applications (for example in Nuclear Science).

The problem you are having is called "pileup" and, as you noted, a direct consequence of not resetting the capacitor.  You need to do what is called "baseline restoration".

One common solution is to add a discharge path as clabacchio suggested.  The stupid simple way to do that would be to put a resistor in parallel with the capacitor you are charging.  Then, the capacitor would discharge based on the RC time constant of the value of the resistor and capacitor you are using.

Doing this makes a brutal tradeoff though, and may not work.  Here's the tradeoff: on one hand you want a small RC time constant so you can have a faster train of pulses.  A small RC means the capacitor will bleed its charge quickly and be ready for another pulse.  On the other hand, you want a large RC time constant so the capacitor holds the peak long enough for you to record the measurement.

If the simple technique I described doesn't work, you can put a transistor switch across the capacitor.  Once you've made the measurement, you pulse this transistor and do a hard reset, clearing the capacitor.  This technique is a bit more complex (especially if you're not using a computer already to acquire the data) but can be effective.  If you're not using a capacitor you need to have a comparator.  One input will be the signal you're measuring and the other will be the threshold.  The output of this comparator can then be connected to the switch.