How do I achieve this kind of color replacement programmatically?

So this i
You'll want to use a ColorMatrix here. The source image is grayscale, all its R, G and B values are equal. Then it is just a matter of replacing black with RGB = (0, 0, 255) for dark blue, white with RGB = (255, 255, 255) to get white. The matrix thus can look like this:
1 0 0 0 0       // not changing red
0 1 0 0 0       // not changing green
0 0 0 0 0       // B = 0
0 0 0 1 0       // not changing alpha
0 0 1 0 1       // B = 255
This sample form reproduces the right side image:
public partial class Form1 : Form {
    public Form1() {
        InitializeComponent();
    }
    private Image mImage;
    protected override void OnPaint(PaintEventArgs e) {
        if (mImage != null) e.Graphics.DrawImage(mImage, Point.Empty);
        base.OnPaint(e);
    }
    private void button1_Click(object sender, EventArgs e) {
        using (var srce = Image.FromFile(@"c:\temp\grayscale.png")) {
            if (mImage != null) mImage.Dispose();
            mImage = new Bitmap(srce.Width, srce.Height);
            float[][] coeff = {
                            new float[] { 1, 0, 0, 0, 0 },
                            new float[] { 0, 1, 0, 0, 0 },
                            new float[] { 0, 0, 0, 0, 0 },
                            new float[] { 0, 0, 0, 1, 0 },
                            new float[] { 0, 0, 1, 0, 1 }};
            ColorMatrix cm = new ColorMatrix(coeff);
            var ia = new ImageAttributes();
            ia.SetColorMatrix(new ColorMatrix(coeff));
            using (var gr = Graphics.FromImage(mImage)) {
                gr.DrawImage(srce, new Rectangle(0, 0, mImage.Width, mImage.Height),
                    0, 0, mImage.Width, mImage.Height, GraphicsUnit.Pixel, ia);
            }
        }
        this.Invalidate();
    }
}
                                                                        If any Android devs end up looking at this, this is what I came up with to gray scale and tint an image using CodesInChaos's formula and the android graphics classes ColorMatrix and ColorMatrixColorFilter.
Thanks for the help!
public static ColorFilter getColorFilter(Context context) {
    final int tint = ContextCompat.getColor(context, R.color.tint);
    final float R = Color.red(tint);
    final float G = Color.green(tint);
    final float B = Color.blue(tint);
    final float Rs = R / 255;
    final float Gs = G / 255;
    final float Bs = B / 255;
    // resultColor = oldColor + (1 - oldColor/255) * tintColor
    final float[] colorTransform = {
            1, -Rs, 0, 0, R,
            1, -Gs, 0, 0, G,
            1, -Bs, 0, 0, B,
            0, 0, 0, 0.9f, 0};
    final ColorMatrix grayMatrix = new ColorMatrix();
    grayMatrix.setSaturation(0f);
    grayMatrix.postConcat(new ColorMatrix(colorTransform));
    return new ColorMatrixColorFilter(grayMatrix);
}
The ColorFilter can then be applied to an ImageView
imageView.setColorFilter(getColorFilter(imageView.getContext()));
                                                                        The formula for calculating the new pixel is:
newColor.R = OldColor;
newColor.G = OldColor;
newColor.B = 255;
Generalizing to arbitrary colors:
I assume you want to map white to white and black to that color. So the formula is newColor = TargetColor + (White - TargetColor) * Input
newColor.R = OldColor + (1 - oldColor / 255.0) * TargetColor.R;
newColor.G = OldColor + (1 - oldColor / 255.0) * TargetColor.G;
newColor.B = OldColor + (1 - oldColor / 255.0) * TargetColor.B;
And then just iterate over the pixels of the image(byte array) and write them to a new RGB array. There are many threads on how to copy an image into a byte array and manipulate it.
This code project article covers this and more: http://www.codeproject.com/KB/GDI-plus/Image_Processing_Lab.aspx
It uses the AForge.NET library to do a Hue filter on an image for a similar effect:
  // create filter
  AForge.Imaging.Filters.HSLFiltering filter =
      new AForge.Imaging.Filters.HSLFiltering( );
  filter.Hue = new IntRange( 340, 20 );
  filter.UpdateHue = false;
  filter.UpdateLuminance = false;
  // apply the filter
  System.Drawing.Bitmap newImage = filter.Apply( image );
                                                                        Depends a lot on what your image format is and what your final format is going to be.
Also depends on what tool you wanna use. You may use:
GDI is quite fast but can be quite cumbersome. You need to change the palette. GDI+ is exposed in .NET and can be slower but easier. OpenCV is great but adds dependency.
(UPDATE)
This code changes the image to blue-scales instead of grey-scales - image format is 32 bit ARGB:
private static unsafe void ChangeColors(string imageFileName)
{
    const int noOfChannels = 4;
    Bitmap img = (Bitmap) Image.FromFile(imageFileName);
    BitmapData data = img.LockBits(new Rectangle(0,0,img.Width, img.Height), ImageLockMode.ReadWrite, img.PixelFormat);
    byte* ptr = (byte*) data.Scan0;
    for (int j = 0; j < data.Height; j++)
    {
        byte* scanPtr = ptr + (j * data.Stride);
        for (int i = 0; i < data.Stride; i++, scanPtr++)
        {
            if (i % noOfChannels == 3)
            { 
                *scanPtr = 255;
                continue;
            }
            if (i % noOfChannels != 0)
            {
                *scanPtr = 0;
            }
        }
    }
    img.UnlockBits(data);
    img.Save(Path.Combine( Path.GetDirectoryName(imageFileName), "result.png"), ImageFormat.Png);
}
                                                                        Easiest would be to use ColorMatrix for processing images, you will even be able to process on fly preview of desired effect - this is how many color filters are made in graphic editing applications. Here and here you can find introductions to color effects using Colormatrix in C#. By using ColorMatrix you can make colorizing filter like you want, as well as sepia, black/white, invert, range, luminosity, contrast, brightness, levels (by multi-pass) etc.
EDIT: Here is example (update - fixed color matrix to shift darker values into blue instead of previous zeroing other than blue parts - and - added 0.5f to blue because on picture above black is changed into 50% blue):
var cm = new ColorMatrix(new float[][]
{
  new float[] {1, 0, 0, 0, 0},
  new float[] {0, 1, 1, 0, 0},
  new float[] {0, 0, 1, 0, 0},
  new float[] {0, 0, 0, 1, 0},
  new float[] {0, 0, 0.5f, 0, 1}
});
var img = Image.FromFile("C:\\img.png");
var ia = new ImageAttributes();
ia.SetColorMatrix(cm);
var bmp = new Bitmap(img.Width, img.Height);
var gfx = Graphics.FromImage(bmp);
var rect = new Rectangle(0, 0, img.Width, img.Height);
gfx.DrawImage(img, rect, 0, 0, img.Width, img.Height, GraphicsUnit.Pixel, ia);
bmp.Save("C:\\processed.png", ImageFormat.Png);