How should I process music play data

Interesting problem. You certainly could use time series methods for this, but you should consider whether you think there is face-validity to song plays following a temporal pattern. From your plot above, I don't see any structure in the variation, but perhaps there is something that an algorithm can pick up. A good starting point would be R's auto.arima function for autoregressive integrated moving average (ARIMA). The ARIMA method is more fully described elsewhere (e.g., here), but an interesting extension to this approach might be using exogenous predictors in your model. For example, does the presence or absence of rain affect the likelihood of some songs being played? My guess would be yes (I love listening to post-rock bands on rainy days), but perhaps that's not true of everyone. This extension to ARIMA is referred to as ARIMAX in the literature. A really nice breakdown of how to handle covariates in your type of analysis can be found on Rob Hyndman's blog.

A second thought is that, with sufficient computational resources, there's an interesting hidden markov model (HMM) problem here. You could potentially model the transition probabilities from one song to another. Presumably there's a high likelihood of transitioning from track 1 to track 2 of a given record, but sometimes people might switch to a different record after listening to a favorite track on an album. The extent to which this is true for different tracks could be a part of this analysis, and you could attempt to model the hidden underlying states that motivate a listener to stay within a record, or to move to a new one.