If you are looking to fit an ARIMA model, first you will have to make sure that the series is stationary. Also check if the variance is stable. If not, I would suggest a ln transformation.
Now important to note, since you have seasonality in your data, your will most like have to take a seasonal difference as well. I would suggest starting with a periodogram, using it to identify the period. From there you can use the ACF and PACF to identify your seasonal part and after that your non-seasonal part of the model. This will give you a SARIMA model. Remember to check that the residuals are white noise before you choose a model.
Forecasting a full year ahead with only two years of data, can prove to be somewhat difficult or rather, it could be inaccurate. Each new forecasted value should be a different week as you stated that each observation is a different week.
Hope this answers your question more or less! You are welcome to ask, give suggestions or make corrections to this.