I expect the best thing to start off with is to get a digital USB ammeter - I have an old one, and I think I paid the princely sum of £5 on eBay for it! Here is one for a similar price. You want something that will do current and cumulative current - the latter will let you measure for an hour or so, to see what the consumption is with the fluctuations ironed out.
Once you have got a bunch of measurements, you can use that to determine what battery you need, given your consumption and expected time between recharges.