Here are some ways that I think can help you get started in play testing your deck.
1. Hang out in a local Hobby shop / Card shop
People who play MTG usually hang out to these places because they look for players like you or maybe look for cards they need from people who visit the shops. A good shop may have several tables and chairs for customers who would like to spend some time with people who have same interests as them and get to play MTG. I'm sure you'll find one near your place.
2. Start by joining an MTG forum
If you're having difficulty looking for a card shop near your place, then you can join a forum that is dedicated to MTG. This is really easy and convenient because you can just post a message and people in your area might have a similar question as you. They are looking for other players to test their decks. If not all but most of the ACTIVE players of MTG will be posting regularly in these forums. So, they will be more than happy to help you out in looking for a place to play test.
In some cases there are no shops near your area. But through MTG forums, I believe you can find a place where you can invite people, same as you, to have some good 'ol Magic discussion and practice to be more competitive.
I know that some players, who are far from the local card shop here in my country, hold their regular MTG meetings (and sometimes hold their SANCTIONED DCI tournaments) at a nearby Starbucks Coffee shop. I'm sure you'll find some fellow MTG players. It's good that you organize with players and find ANY place with some tables and chairs and make that your weekly hang-out (hopefully the owner of that place allows you to play there. At least you can ask them too.).
Not available during Friday nights? Not to worry. FNM is not the only place and time to practice with your deck! I hope what I've stated before, will help you get around this problem. I also would like to suggest that you can try other formats. Because from my experience and understanding, FNM usually host the Standard format of the game. So, by simply changing your format to Extended, Legacy or Vintage. You can test with your fellow format MTG players! :D Finding people will be easy.
So, in short, to find people to playtest with you. You need to hang out to a card shop or search through the internet by joining an MTG forum. :D
Good luck on your search.