There's lots of misunderstandings here, but here's the gist of it.

What you want to do is a perfect application of an Arduino. No need to buy a coulomb counter, instead, you should **make one** with your Arduino.

> I would then add this amp/second value to a variable. 

First correction, it's an **amp\*second**, NOT an amp/second. You are multiplying amps by the number of seconds that current value existed. 

amp\*seconds / 3600 = amp\*hours. Read about that here: https://en.wikipedia.org/wiki/Ampere_hour. Multiply that by the _voltage_ over that same time period and you've got amps\*hours\*volts = (amps\*volts)\*hours = Watts\*hours = units of *energy*! Remember, your house electricity it purchased in units of kilowatt hours, which is kilowatts\*hours, or kWhr. And in engineering units a [**joule**](https://en.wikipedia.org/wiki/Joule) is  the standard unit of energy, and it is simply Watts\*seconds. So, you can convert kWhr to joules like this, where units are in square brackets `[]`.

    Joules = kWhr * 1000 [Whr/kWhr] * 3600 [sec/hr] = [Watts*sec] = [Joules]

So, 1 kWhr x 1000 x 3600 = 3600000 Joules = 3600kJ

> I would also increment another variable for each second.

That doesn't make any sense. Don't do that. You've already captured the seconds by adding your new bit of amps*seconds to your old sum of it. 

DO account for voltage too! What you're really after is *energy*, which can be expressed in units of joules or kWhr, as already described above.

> Is it too much a stretch to **poll** the amperage every second and call it **amp\*second**? 

(my corrections in bold)

No, absolutely not! That's *exactly* what an amp*second is! It's the current, amps, times the number of seconds that current existed. That's a coulomb counter!

> How can I make this more accurate? 

This is the part where the engineer and programmer (you) determines the quality of the result. 

1. Don't assume 1 second elapsed because you *told it to*. *Measure it* for each and every measurement!
1. The smaller the time interval, the more accurate, since it allows you to capture the small current fluctuations, so use a *smaller time slice* than 1 second. I'd start with 10ms. 
1. Measure voltage AND current AND actual time elapsed on this loop as close to the same time as possible.
1. Do NOT use blocking techniques, such as `delay()` or `delayMicroseconds()`. 
1. What you're doing when you're multiplying a new amp reading x seconds, and adding it to the previous sum is called [*numerical integration*][1]. You're simply numerically integrating, or summing, the area under the Amp (y-axis)-time (x-axis) curve, or, if you're doing what I'm saying to do intead: the Watt (y-axis)-time (x-axis) curve, where the area under the curve is the total energy collected. **Don't do rectangular integration, do *trapezoidal* integration following the *trapezoidal rule*! It's more accurate!**
1. Calibrate your clock! An Arduino's clock can be off by a couple percent. Figure out how much it's off by comparing its clock output over, say, 10 minutes, and comparing it to a better clock source, say...an internet time server displaying the atomic clock in Colorado or wherever it is. Then, multiply each measurement by a scaling factor to fix it! 
1. On the same lines as above, ideally you'd also compensate in real-time for temperature, as the clock frequency drifts with temperature. 



  [1]: https://en.wikipedia.org/wiki/Numerical_integration