Chorus and delay effects are useful for getting this kind of sound. Most daws come with a stock plugin for each of these, but there's many free/cheap ones that can be used as well, for different sounds. You can also use things like vocoders to big, full, robot-like choir sounds, but they tend to be neither free nor cheap unfortunately.
One thing that's worth noting about using the same voice wav file and panning it hard left+right alongside the centre - if it is the same file, with no changes, it will make the vocal sound louder, but it won't make it sound fuller and get this choir effect. With human singers, you just get the singer to record a bunch of different takes, as each one will naturally vary and can be used pretty much straight away for choruses. With vocaloids, you have to make each of the takes vary with tuning. Changing parameters globally on the backups (breathiness, gender, etc) can be a good way to start differentiating them. You also wouldn't want them to have the exact same tuning nuances as the lead vocal - in general, backup vocals are more pitch corrected and less dynamic, so getting rid of big pitch/dynamic edits will help differentiate them as well. Lowering the brightness compared to the main vocal will also help them sit behind it and give them a different tone.
Of course, even with editing the voice in this way, Vocaloid is still drawing from the same recorded sample, which is where delay/chorus plugins come in. They shift the timing of the vocals, so the same samples aren't playing at the exact same time and just boosting volume instead of filling out the sound. They also can add some artificial variance to the sound - I don't know exactly how it works, but they can add subtle pitch/dynamic fluctuations to the sound to help differentiate it. The timing part of delay can be achieved by dropping the file in and manually dragging it back on the timeline, but it won't add the pitch/dynamic variations this way.
In terms of how to add this to a song - start by rendering your main vocal, then render your backup vocals with tweaked parameters to differentiate them (try make them different to each other, too). Drop them all in your daw. Don't put any delay on your main vocal directly. Add delay to your backups, and make it 100% wet, so none of the same-timing-as-lead sound is coming through. Tweak the delay time to taste - for chorus about 30ms is a good place to start, but you can even go as high as 100+ depending on the song tempo/the kind of sound you want. You want them all to be different from each other. There may be other controls on the plugin such as lfo or modulation depth - these control the subtle variance of the delayed file. I'd recommend making these slightly different for each double. There may also be controls for spread, feedback, etc. These are more for delay effects than chorus effects, but you can play with them to see what they do and see if they help you get the sound you want. Spread could be v helpful