If you have independent data, you will want to either run an ARIMAX model or a regression with ARIMA errors. There is a difference. Parameter estimates are easier to interpret in regressions with ARIMA errors, which R's
auto.arima() and other functions fit.
If you have meteorological data, I'd expect intra-daily seasonality. With data in 5 minute buckets, this translates to a season length of 288, which is quite a lot. Fitting an ARIMA model to long-season data will take a while. In R, you can simulate four weeks' worth of white noise with a seasonal period of 288 and then check how long fitting an ARIMA model will take like this:
foo <- ts(rnorm(4*7*24*12),frequency=24*12)
I'd love to tell you how long this took on my machine, but after five minutes, it's still running, and I'm getting bored. It might be better not to let
auto.arima() choose the seasonality, but to force seasonality by setting the parameter
D=1 - this might speed up matters a bit.
Other software may have similar possibilities of prespecifying the seasonality, and of including covariates.