How can I make lemon and chilli liquid to have with hummus?
In Israel I have often seen hummus/falafel/thina served with a hot sauce called skhug, I have mostly seen the green variety (skhug yarok), which is a sauce made of fresh herbs, garlic, chili, lemon juice, garlic, olive oil, and some spices.
Hummus is often just served with thina on the side and with olive oil, but there is a lot of variety ... I have seen sauces based on olive oil and lemon juice, similar to what you describe (although not biter ... but the bitterness could also come from blending the olive oil). But I never saw it being prepared in front of me or it being called a particular name, but it looked a little like a thin vinaigrette.
Using my extremely limited hebrew knowledge and google I found this recipe.

The ingredients translate to:

2 hot chilies
3-4 cloves of garlic
3 tbsp olive oil
juice of half a lemon
salt
additional recomendation:
1 tsp cumin
sugar
Preparation: Put all ingredients into the food processor and turn into the sauce. Taste and adjust spices/oil/lemon juice to taste. And add sugar if too sour to balance the flavor.

You can let sit a little for the flavors to combine ... and strain out remaining bits if desired.
I am not sure if this is the sauce that you had ... but it could be very similar. Anyway ... it sounds tasty.