How to Create a Custom Typography Logo?

If you have access to Lynda.com, Von Glitschka has some excellent videos on working with the pen tool.

Von Glitschka — Online Courses, Classes, Training, Tutorials on Lynda

IF you want a more organic rough look exactly as drawn, you can image trace. Make sure to check ignore white.

You can then object >> expand to get to the vector points

For a smoother look you can tweak the settings.

If you're new to Illustrator, the Pen tool might be tricky at first. You can also use the pencil tool (N), which lets you redraw segments. You can smooth out curves with the smooth tool. As a path, it can be widened with the Width tool (Shift + W).

You can also try Image Trace. Try the Black and White setting. Then expand it. You can use Object > Path > Simplify and adjust the Curve Precision to smooth it out. This Image Trace setting makes it an object with a black fill, so if you want to adjust the width, you'll need to use the Direct Selection Tool (A) to adjust the path that outlines the shape.

I'd use the pen tool for this. Click for sharp corners, click and drag for rounded areas. Then use a stroke profile and rounded corners (or a brush style) to make it look more freehandish. Remember that fewer points with make a smoother line.