you need to do a lot of experiments, listen to a lot of songs and observe the backing vocals.
on top of that, you must have a strong music theory in terms of harmony (chords), or else you'll stuck when there're chords you don't know.
and most importantly, backing vocal need well trained ear to listen to the harmony, you'll have to tell what kind of chords are they by just listening or it'll be time consuming to analyze them on piano.
a few approach are 3rds, 5ths, 6ths, and octaves. sometimes 4ths are also nice. dig out your SATB of the ABRSM Theory but it's not classical so breaking the rules is OK given that it sounds good.

btw, nobody in the profesional field ever want to use software to do backing vocals, they'll never cut it.
i agree with you little ice. But not all in the professional industry doesn't use the harmonizer and pitch shifter. They need to use it if they have to. But for this case from TS, he can use both plugins and the piano. It doesn't harm to use the plugins anyway, Why don't we give TS a try rite little ice?