There are at least two ways. The first is by taking part in political activities. The second way is to do non-political things with political intent. You can choose to do one or the other, or both, or neither. Many disagreements among faithful Christians stem from a misunderstanding of these options.
Political action versus intent
Some actions are inherently political. These include voting, campaigning, and demonstrating for a cause, like marching in Washington, DC. The actions are aimed at accomplishing a political outcome, and they're done within the rules of a political "game" (e.g., vote-getting or applying legislative pressure).
Non-political activities are, by contrast, things that aren't specifically aimed at political outcomes and that aren't performed under such rules. Everything from mowing the lawn to watching TV to breastfeeding—even paying taxes or driving the speed limit. They're not necessarily insulated, asocial actions, but they're not inherently political, either. If they're done in public or for a public purpose, however, they can transmit some level of political intent.
Our duty to be political
People are naturally political, and being political is a good thing. So much so that we have a duty to be political. Concern for our community and our state is a noble feature of our humanity. Christians, too, must "render to Caesar the things that are Caesar's." Yet the natural charge to be political doesn't boil down to a requirement to be political in a certain way. And for Christians, it can't mean being political in a way that conflicts with the first, higher duty to render "to God the things that are God's."
Political duty arises from our personal nature, and good politics coincides with human flourishing. Some types of actions are better suited than others, however, to discern where exactly that occurs. Political duty more closely aligns with rendering to God what is God's than it does with participating in explicitly political activities. To put it simply, as an example, voting is good when our vote renders not merely what is due to Caesar (i.e., civic performance), but also when it renders what is due to God (i.e., an act of fidelity driven by moral certainty).
Not a zero-sum game
Politics isn't a zero-sum game—at least, not always. Voting is zero-sum: There aren't more votes than what's available. But voting isn't everything, and since political action moves at different speeds and in different ways, the pie of politics can grow.
It's even possible that actions that conflict on the surface help create a richer, better political atmosphere for real things like human life, family integrity, economic stability, national defense, etc. Working from the "inside out"—moving from politics as a duty to God toward a more civic expression—drives this. For example, when Christians personally and carefully reject explicit political engagement on a particular point (e.g., voting for a major candidate) in order to preserve some principle, it doesn't necessarily undermine the positive effects produced by those who, in good conscience, can engage. Arguing that no one should perform overtly political acts would almost always be irresponsible. But a tempered, personal witness of discerning the value of principle versus pragmatism might not only produce good fruit for the individual, but can also grow the entire political pie. A short-term, practical conflict can expand a long-term, substantive point of view.
The people who direct political success will always be those who help form others' consciences to make specific, practical choices (good or bad). For Christians to be politically significant, we must do more than just participate in the political activities that are handed to us. We must set a political agenda through non-political, personal choices made with the health of society and our state in mind.