https://www.asus.com/Motherboards/P7H55M_LX/specifications/
You shouldn't really have any trouble with any GPUs out there, but given your platform & budget I would go with an RX 570. For your price range, it's the best card for 1080p games, especially since I doubt you have a 144Hz monitor. That's also a huge boost in graphics capability, as you'd essentially move from a 31st-tier GPU to a 4th-tier GPU (http://www.tomshardware.com/reviews/gpu-hierarchy,4388.html).
That being said...your Core i3 is going to possibly hold you back in some games (depending on what you're playing). Unfortunately, there aren't a whole lot of options available for you. Your motherboard can handle i5/i7 CPUs (https://www.asus.com/Motherboards/P7H55M_LX/HelpDesk_CPU/), but the problem is two-fold:
1. Cost. I couldn't find any Lynnfield core i7 chips available on PCPartPicker, & 2 of the 4 core i5 chips they listed were more than your overall budget (https://pcpartpicker.com/products/cpu/#m=21&s=12,13&f=12). That would severely limit your budget for a GPU, although you could still squeeze a GTX 1050 in (MSI GTX 1050 2GB OC, https://pcpartpicker.com/product/gXnG3C/msi-geforce-gtx-1050-2gb-video-card-gtx-1050-2g-oc).
2. Performance. Unlike later models, the Clarkdale core i5 chips (i5-650/660/670/680) are 2C/4T chips just like the core i3 you already have; their only benefit is the slightly higher clocks (3.2 to 3.6GHz base vs. 3.07GHz) & having the Turbo boost (boost up from 3.3 to 3.7GHz for both cores or from 3.4 to 3.8GHz using only 1 core, depending on the model). With only a little bit of performance improvement, I'm not surprised that the core i5 chips are only ranked 1 tier higher than your current i3 (http://www.tomshardware.com/reviews/cpu-hierarchy,4312.html)...& usually you want to improve at least 2 tiers (if not more) for a CPU upgrade. It's too bad that PCPartPicker didn't show any Lynnfield core i5 chips available (i5-750/760), which are full 4C/4T chips; even though they have slower clocks (2.67/2.8GHz base; 2.77/3.07GHz Turbo for 3 or 4/1 or 2 cores on i5-750, 2.9/3.2GHz Turbo for 3 or 4/1 or 2 cores on i5-760), they were ranked 2 tiers higher than your i3-540.
In any case, I wouldn't worry about replacing your CPU. The options are way too limited for the price you'd pay, & you'll get a lot more mileage from a GPU upgrade. Plus, if you start saving up to eventually replace your CPU/Motherboard/RAM, you'll be able to use the new GPU with it.