If you really want to stick to that budget, any old tarp will do. They're heavy, bulky & loud in the wind, but they'll keep the rain off of you.You should be able to find one that's big enough for relatively cheap, especially if you're looking on Amazon.

I actually just picked up a cheap "hammock" tarp from Amazon to bring with as a wind wall/part of a tarp city for our BWCA trip. It was like 30 maybe? Comes with string for tie downs. It's certainly not high quality and is much bigger than the Warbonnet SuperFly I use for my hammock, but again, it will keep things dry.

https://www.amazon.com/gp/product/B08TQS4HRB/ref=ppx_yo_dt_b_asin_title_o00_s00?ie=UTF8&psc=1 this is the one I bought to use a as a crude rain/wind shelter.

Again, these tarps are like 1/4 the price of what some other tarps cost. Just like with anything else, you get what you pay for. If this is your first time, something cheap like the tarp linked above might be enough to get you hooked on hammock camping. I went on many trips with an old-school diamond rainfly & have had so much fun expanding & exploring new gear. Give it a try!

ETA: you mentioned this is a backpacking trip. Weight really matters here, and that's where the price differences come in. Also, hammock camping can take a bit of practice to get it dialed in right & there is nothing worse than waking up wet & cold because something wasn't set up right. It can happen to anyone & that can completely ruin a trip. I'd advise doing a test run or two before relying on this as your shelter in the wilderness. Even after years I made a mistake that could have been deadly had I been alone/unable to get warm again. Just my two cents!