There are quite a few things you can do, and several exercises that I use with my ESL clients for doing just this.
For example, three of the top ones:
- First, learn to chunk your English properly (this is the number one big thing; native speakers don’t speak in words and rules – they speak in chunks and pronounce language in chunks. Until you learn to articulate your English in the same way, you’ll always sound flat and monotonous — if you’re not sure what “chunking” is or how to do it, have a look at the free training I created here).
- Learn to vary the speed of your voice for effect (it’s not true native speakers speaker “fast” – we speed up and slow down all the time.
- Practise extremes: one of my favourite is reading aloud in an exaggerated, extreme way. You wouldn’t speak like this in an actual conversation, but by exaggerating in your practise, displaying emotion, anger, excitement, drama etc. in your speech will feel more comfortable.
There are more, but that’ll get you started.