What should I do if I want to do a PhD at a particular lab but cannot secure a scholarship? I was able to make a good impression on a potential supervisor and he was willing to take me in provided I manage to secure a funding. We both hoped I could get the scholarship from PhD programme but I was not selected.

Now, I still want to join his lab but I was strongly discouraged from doing a totally non-funded PhD. I could wait and apply next year but I don't want a year break. I also don't plan doing a Masters degree as I got plenty of experience from my industrial placement year. Plus, I hope to save paying international tuition fees.

I talked with my university tutor and other researchers that I know and they encouraged me to contact the supervisor and try any way possible to work with him. I could propose working voluntarily during the summer (start the project maybe?) and ask for a technician/assistant position (short contract, 1-2 years) with the hope of him applying for grants when I'm already there, showing my skills etc? That's what I was advised to do.

I wonder how should I phrase that in the email. I don't want to write too much but I still need to convey everything. I guess, I'm not too good at writing and I don't want to ruin any chances I have with him.
Talking to your prospective advisor is the best option as they should know what other funding opportunities there are. If you demonstrate determination to study a PhD with them and enthusiasm for research they will most likely try harder to find you funding and support you in the mean time. Try to develop ideas for the PhD, think about what you really want to research. Volunteering or working temporarily for the advisor could be a way in to potentially get a publication. If you can get your name on a publication, you will be much more likely to secure funding. But I suggest you call the professor instead of emailing, especially if you are not comfortable writing the email. 

As @Sayan mentioned, making multiple applications to different laboratories is also a good idea. International tuition is often crippling, but there might be better funding opportunities in another country which could take care of that. Letting the potential supervisor know you are doing this will further demonstrate your enthusiasm for research, and taking up an offer elsewhere does not mean you will never work with your first choice.