In a word, no. Brita-type filters are designed to take out waterborne contaminants such as chemicals, and to remove sediment. They aren't at all intended to "purify" water or remove biological nasties such as giardia.
So, get a filter that's designed for backcountry and camping use. Try an MSR Hyperflow ($100)—the new design has an extremely high flow rate (three liters per minute) and attaches easily to a Nalgene-type bottle. It takes out protozoa, bacteria, and sediment. It won't get viruses—for that you need a post-filtering add-in such as MSR's SweetWater Purifier Solution ($10). If you're camping with a group of people, you might try another MSR product, the Autoflow Microfilter ($100).
It's a gravity-fed system. You simply fill a compatible water bag, attach the filter, and hang it up. Flow rate is close to two liters per minute, and it does everything that the Hyperflow does. You can have water for four in minutes without a single pump-stroke.
The Katadyn Base Camp Water Filter ($70) works the same way. And of course Katadyn makes hand-held filters as well, such as the ruggedly built Katadyn Pocket Water Filter ($290). For filtered water on the go, you also can use an individual bottle filter system like Sawyer's Four-Way Filter ($55), which comes with a bottle that holds 32 fluid ounces. Its filter can be used with the bottle, so you can sip any time. Or use it to make a gravity filter system, or even attach it to a faucet. Best for a single user, however.