Optimizing paint code

Magic numbers

You have a lot of magic numbers in your code. Consider to express them as meaningful constants. ( 4000, 112, 15, 16, 640, 12, 8f, 11f)

Style

Be consistent with your coding style.

string s = System.Text.Encoding.ASCII.GetString(_screenMemory, i, 1);
var pf = new PointF(xLoc, yLoc);  

at the first line you explicitly declare s as string also it is quite obvious by the right hand side that it is a string. On the second line you use var as expected.

There is nothing wrong with declaring implict by using the type, but here you should have either used var also or instead of var PointF.

Naming

Use meaningful names for variables. Mr.Maintainer will thank you for doing so. s and pf are poor names.

Using statement

Instead of creating and disposing the Graphics you should enclose this inside a using statement, so in case of an exception the Graphics object is properly disposed.

Declaring variables

Variables should be declared as nearly as possible of their use ( here: the brushes ).

Refactoring

This won't do anything regarding speed, if the bottleneck are the FillRectangle() and DrawString() methods.

There is no sense in entering the loop or doing anything if _resetInvoked==true. So using a guard condition to return early is the way to go.

private void Screen_Paint(object sender, PaintEventArgs e)
{

    if (_resetInvoked) { return; }  
    _resetInvoked = true;

    using (Graphics bmpGraphics = Graphics.FromImage(_screenBitmap))
    {

        int xLoc = 0;
        int yLoc = 0;

        bmpGraphics.Clear(Color.Black);

        for (int i = 0; i < 4000; i += 2)
        {

            if (((xLoc % 640) == 0) && (xLoc != 0))
            {
                yLoc += 12;
                xLoc = 0;
            }

            var bgBit = (byte) ((_screenMemory[i + 1]) & 112);
            SolidBrush bgBrush = _backgroundBrushesArray[bgBit/16];
            bmpGraphics.FillRectangle(bgBrush, xLoc + 2, yLoc + 3, 8f, 11f);

            var fgBit = (byte)((_screenMemory[i + 1]) & 15);
            SolidBrush fgBrush = _foregroundBrushesArray[fgBit];

            var currentText = System.Text.Encoding.ASCII.GetString(_screenMemory, i, 1);
            var currentPoint = new PointF(xLoc, yLoc);
            bmpGraphics.DrawString(currentText, _renderFont, fgBrush, currentPoint);

            xLoc += 8;
        }

    }

    e.Graphics.DrawImage(_screenBitmap, new Point(0, 0));
    _resetInvoked = false;
}