You have several options on how you could receive the httpMethod in Java. One of the easiest would be to rename 'http-method' to 'httpMethod' in your Integration Request in API Gateway and then use the RequestHandler Interface for your Lambda handler which will marshal your JSON directly to a Java Object:
package example;
import com.amazonaws.services.lambda.runtime.RequestHandler;
import com.amazonaws.services.lambda.runtime.Context; 
public class Hello implements RequestHandler<PojoRequest, PojoResponse> {
    public PojoResponse handleRequest(PojoRequest request, Context context) {
        System.out.println(String.format("HTTP method is %s.", request.getHttpMethod()));
        return new PojoResponse();
    }
}
Then you can create whatever Pojo you want to be the request, for example:
package example;
public class PojoRequest {
    private String firstName;
    private String lastName;
    private String httpMethod;
    public String getFirstName() {
        return firstName;
    }
    public void setFirstName(String firstName) {
        this.firstName = firstName;
    }
    public String getLastName() {
        return lastName;
    }
    public void setLastName(String lastName) {
        this.lastName = lastName;
    }
    public String getHttpMethod() {
        return httpMethod;
    }
    public void setHttpMethod(String httpMethod) {
        this.httpMethod = httpMethod;
    }
}
See: http://docs.aws.amazon.com/lambda/latest/dg/java-handler-using-predefined-interfaces.html