The promise of lightning network is fast and cheap transaction when doing micro payment.
The promise of lightning network is implemented; it's no longer a promise comparing with lots of altcoins that pledge to solve the scaling issue.
I want to know, with near approximation, what can be the gas fee for making a payment of $5 using ligtning network?
You mean the transaction fee?
Well, it depends. If you want to transact once and never use the LN again, then you'll have to pay for two on-chain transactions; one for opening the channel and one for closing it. In other words, if you want to transact only once, you have no reason to use the LN. Just broadcast an on-chain transaction instead. The lightning network exists to achieve transacting micro amounts with the lowest fee possible.
It also allows you to spend dust amounts in constrast with the main layer.If you want to start using it and being keen on it, then you'll realize how cheap it really is. Each transaction I've made costed 1 satoshi. ONE!