Create a quad, prepare it for drawing, and edit some pixels. Remember use the Sprite Material to get the desired effect. This is just a barebones example on how to display and edit an arbitrary Texture2D.

RandomSprite.cs

using UnityEngine;

[RequireComponent(typeof(MeshRenderer)), RequireComponent(typeof(MeshFilter))]
public class RandomSprite : MonoBehaviour
{

    private static float quadScale = 1f; // Size of our Quad in World Units
    private static Vector3[] quadVertices = new Vector3[] { new Vector3(0f, 0f, 0f),
                                                            new Vector3(quadScale, 0f, 0f),
                                                            new Vector3(0f, quadScale, 0f),
                                                            new Vector3(quadScale, quadScale, 0f)};

    private static int[] quadIndices = new int[] { 0, 3, 1, 0, 2, 3 };


    private static Vector2[] quadUVs = new Vector2[] { new Vector2(0f, 0f),
                                                       new Vector2(1f, 0f),
                                                       new Vector2(0f, 1f),
                                                       new Vector2(1f, 1f)};

    private int width = 16; // The dimensions of our Texture in Pixels
    private int height = 16;

    MeshRenderer myMeshRenderer;
    MeshFilter myMeshFilter;
    Texture2D myTexture;

    void Start()
    {
        // Get out references to our components
        myMeshRenderer = GetComponent<MeshRenderer>();
        myMeshFilter = GetComponent<MeshFilter>();

        // Create a texture for this GameObject
        myTexture = new Texture2D(width, height);
        myTexture.filterMode = FilterMode.Point; // Don't blur our texture

        myMeshRenderer.material.mainTexture = myTexture; // Apply our texture to our renderer

        // Create a new mesh (so we have a shape to render!)
        Mesh tMesh = new Mesh();
        tMesh.vertices = quadVertices;
        tMesh.triangles = quadIndices;
        tMesh.uv = quadUVs;

        myMeshFilter.mesh = tMesh; // Apply our mesh to our filter

        // Clear our texture
        for (int i = 0; i < width; i ++)
        {
            for (int j = 0; j < height; j++)
            {
                SetPixNoApply(i, j, new Color(0f, 0f, 0f, 0f));
            }
        }
        ApplyTexture(); // Apply our changes

        // Make some other changes to our texture
        SetPixNoApply(1, 1, Color.red);
        SetPixNoApply(2, 1, new Color(0f, 0f, 1f));
        SetPixNoApply(3, 1, new Color(0f, 1f, 0f));
        SetPixNoApply(4, 1, Color.red);
        ApplyTexture(); // Apply those changes too
    }

    // This method is faster but you must call ApplyTexture() after calling this
    void SetPixNoApply(int x, int y, Color c)
    {
        if (myTexture != null)
        {
            myTexture.SetPixel(x, y, c);
        }
    }

    // This method applies changes to our texture
    void ApplyTexture()
    {
        if (myTexture != null)
        {
            myTexture.Apply();
        }
    }

    // This method is slower because it calls Apply() after every pixel change
    // Only use this when changing just a few pixels at a time
    // If you are changing more than one pixel at a time, you should use
    // SetPixNoApply for all pixels, then call ApplyTexture() instead
    void SetPix(int x, int y, Color c)
    {
        SetPixNoApply(x, y, c);
        ApplyTexture();
    }

}