You need an event to act as the catalyst for your code being invoked. This can be the constructor of the page or an action attribute. Then, assign the value as normal during your action event, and it will update the inputfield. For example:

// Controller
public with sharing class TestController {
  public string Country { get; set; }
  public string ExchangeRate { get; set; }

  public void updateExchangeRate() {
    Http gate = new Http();
    HttpRequest req = new HttpRequest();
    req.setEndpoint('http://www.yoursite.com/path/to/service/');
    req.setMethod('get');
    HttpResponse res = gate.send(req);
    ExchangeRate = res.getBody();
  }
}

// page
<apex:page controller="TestController">
 <apex:form id="theform">
  <apex:pageBlock>
   <apex:pageBlockSection>
    <apex:inputText value="{!Country}">
      <apex:actionSupport event="onchange" action="{!updateExchangeRate}" reRender="theForm"/>
    </apex:inputText>
    <apex:inputText value="{!ExchangeRate}"/>
   </apex:pageBlockSection>
  </apex:pageBlock>
 </apex:form>
</apex:page>

Of course, it would be slightly more complex (maybe you want a dropdown, or maybe it runs off of an object, etc), but the logic will be essentially the same. Attach an actionSupport, or use a commandButton, commandLink, or actionFunction to cause your code to be executed. Assign the results to a binding variable, and your results will automatically appear when the page refreshes.