One of the most distinctive elements of Flume's sound is chopped up sounds, especially vocals. You could either record your own vocals, find a guest vocalist, or search through sample packs for good vocal samples. Flume uses sample packs a fair bit and says he's picky about choosing samples for his productions:
Also, I download a lot of sample packs. So I go through thousands and thousands of kick drums and snare drums to find the right sound. I’m very particular about my sounds. They all have to be unique. I don’t want to use standard sounds, I want to find weird ones.
I'm using a sample from Samplephonics "Kruisemode: Liquid RnB" that was used by Flume in the song Numb & Getting Colder from Skin. Here's the sample over my track with no cuts, I've repitched it to fit the key of my song, in this case 2 semitones down to C Major. It sounds good but it doesn't flow with my track particularly well, especially towards the end of the sample.
To get the sample to fit with your track better, use your DAW's tools to cut, paste, duplicate and move the sound around the beat and add rhythm to it. Reversing portions of the sample is also a great way to get a slightly surreal effect. To explore this technique even further research repitching your samples and applying them to a keyboard to use pitchbend and portamento. Listen to my chopped up version, I'm a big fan of duplicating small sections to get a stuttering effect.
Make sure to root through some sample packs to find your own interesting samples, they don't necessarily have to be similar to the one that I used, it's how you use the sample that will define your sound.
Remember to check out the full article, with synth patches, production tips and a free Ableton Live project file here.