You're not very likely to find one that'll game well at all, certainly not for BF4. Around $750 - $850 would be a "budget" gaming laptop. I really, really recommend you save up for a while if you want a really good budget gaming laptop because Lenovo's Y series simply can't be beat in terms of price : performance value.
Your only hope for a laptop that can game at $500 (you won't even find anything like that for $300) will have an AMD APU. Basically, APUs are CPUs with integrated GPUs. The good thing is, they're cheap and AMD GPU's are far more efficient than the crap offered by Intel. The bad thing is, the budget CPUs that AMD makes as a part of their APU series are nothing in comparison to the quality of an Intel CPU, but overall unless you find a really good refurbished deal or something on Craigslist, there's really zero chance that you'll get anything powerful enough for what you're looking for. In fact, the best you could probably get at that price range (assuming it's new) might get 30 fps at the lowest settings at Battlefield 4 if you're lucky.
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