As far as I know Drow Sign Language does not officially exist in 5th. is this correct?

I want to propose to my group which includes a drow  that we use hand signals to communicate . If Drow Sign Language exists in 5th what are the rules/guidelines for learning it ?

I am sure groups had some kind of systems that used hand signals and I am curious how it worked out.  Was it military hand signals or something else?

The only time ive used it was when my friend and I were playing characters that had a loy of experience fighting and hunting together.

One house rule I've heard is to treat it as a language that can be learned with downtime.

Sign Language is simply that: a language. The rules for it are the same as any other language, be it official or homebrew.

You could select it as part of your race or background, as part of the Linguistic feat or from downtime (rules for this are in Xanathar's Guide to Everything).

D&D Beyond does not yet support homebrew languages in the builder but you can add it directly on the sheet. If you would be taking the language as part of a feat, background etc, then just leave that option blank and add it to the sheet directly. If adding from downtime learning then just add it to the sheet. On the sheet under languages is a small gear icon, click this, click customise and you have the option to add a custom language: just type in the language name and you're done.

Agreed.  Sign language is as complex as any language so just treat it like a normal language in your game.  There is nothing official in 5e for dealing specifically with languages for the deaf to my knowledge but that shouldn't stop you from using such a language in your game.  

That being said, if all you need is simple hand signals similar to what the military uses and don't need to convey complex ideas, then just use whatever signals you want.   They would certainly be easier to learn.  I'm guessing anyone who has ever watched an action movie knows at least a few of those signals.  An actual sign language is much more complex.  It's the difference between waving and telling your life story.  

Of course, different restrictions apply to the language such as the Blindness status effect instead of the Deafened status effect and having access to the Somatic components... possibly two hands and not just one hand and possibly one or both arms free as well.

I know a group that keeps retconning "signs we've definitely worked out beforehand" to silently communicate strategy (even though one of them has a Message cantrip - it's funnier the other way). (DM: "Your hands are bound." Player: "I'm saying it with my eyes. *player wiggles eyebrows and darts eyes everywhere*" DM (highly amused): "That's fair.") I suspect that, one day, their DM is going to make it impossible for them to see each other just to rob them of the facial sign language they apparently invented while travelling together.