When you do something in a determined or deliberate way, you do it purposefully. If you finished your brother's favorite cereal yesterday, he may purposefully shake the last of your favorite kind into his bowl this morning.
Acting purposefully means doing something on purpose, or in a way that demonstrates your sense of resolve. Soldiers trudging purposefully into battle are determined and have their eyes wide open — they know what they're doing. If you purposefully avoid talking about politics with your family during the holidays, that means you intentionally steer clear of controversial topics.