Forecasting time series: Method Selection
You can try two different approaches:

1) Kalman filter, the method is battle-tested and has proven useful in many areas.

Resources:


Understanding the Basis of the Kalman Filter
kalman filter in pictures


2) Recurrent Neural Networks, the LSTM and GRU architectures are particularly interesting for time series predictions.

Resources:


RNN effectiveness
Understanding LSTMs


To do regression and predict future data points, you would need to build a training dataset consisting of a sequence of events. Let's say a value $x$ for every timestamp $t$.

Your data seems to have 1 dimension, so both the network input layer and the output layer would consist of 1 unit. You would then train your model to predict $(x_{t+1})$ given $(x_{t})$.

Let $M$ be our trained model and let's say you want to forecast a data point at time $k$ and you know the current value at time $t$.

$M(x_{t}) = (x_{t+1})$

$t = t+1$

$M(x_{t}) = (x_{t+1})$

$...$ increment $t$ and keep predicting until $t+1 = k-1$

$M(x_{t+1}) = (x_{k})$

Put in pictures this corresponds to:


(picture from Udacity lecture about Deep Learning)