New CPU for LGA 775

Within your budget you could purchase a new motherboard and an i3 processor, which will beat any Core 2 Quad you decide to purchase; thus reducing your bottleneck.

I've noticed your existing motherboard has built-in WiFi, supports SATA and IDE/PATA in addition to DDR3 RAM. In order to upgrade to a new motherboard on a newer socket which supports i3, i5, i7 and Xeon's you should use your existing components (hence not exceeding your budget). However if you're using any of the PATA/IDE connectors, you'll need to purchase SATA alternatives for the new motherboard; however SATA is the successor to IDE and you should be using that anyway, as your current motherboard does support it.

Next we need to consider your current RAM, whether it will be compatible with the new motherboard or not. Is your DDR3 RAM frequency at 800, 1066 or 1333MHz? If it's 800MHz, it won't be compatible with a new board whereas if it's 1333 or 1066MHz, then you're safe. Since you have 5GB RAM (an unusual number), I'm assuming you have 2x2GB modules and 1x1GB; hence I've selected a motherboard with 4 RAM slots.

The build below within your budget is an ideal upgrade for you. The motherboard supports all the latest processors (and future Broadwell processors too), I have included a WiFi adapter to make up for the non-integrated WiFi and an i3-4150 CPU. An i3 is ideal for you as Battlefield utilises the Hyper Threading hence should run a lot smoother and provide better performance.

PCPartPicker part list / Price breakdown by merchant

CPU: Intel Core i3-4150 3.5GHz Dual-Core Processor ($114.99 @ NCIX US)
Motherboard: ASRock H97M PRO4 Micro ATX LGA1150 Motherboard ($84.98 @ Newegg)
Wireless Network Adapter: Encore ENLWI-N3 802.11b/g/n PCI Wi-Fi Adapter ($10.99 @ Amazon)
Total: $210.96
Prices include shipping, taxes, and discounts when available

I would recommend the upgrade above, however if you desire to stick to your current 775 motherboard, I'd look for the Q8300 processor to upgrade to. The Q8300 is the Core 2 Quad (Q8300) at 2.5GHz, evidently a quad-core processor; the Q8300 provides better performance than the Q6600 yet around the same price. You can pick up a used Q8300 on eBay for slightly less than $40; you can't pick up these processors new since they're outdated.

Regardless I'd do with the i3 rather than the Core 2 Quad, but it's completely your call.

All the best.