How would I go about making a bevel but making the bevel weight larger as its transitions from 1 to 2 to 3 to 4. As you can see below the bevel is only one weight. I need it to gradually and evenly get more weight as it goes from 1 to 4
It is hard to do it accurately in Blender, but you could approximate the result using the help of Proportional Edit
Just bevel regularly like you already did, but make sure you leave the 3D Cursor at the exact position of the original unbeveled corner vertex on 1.
If you already beveled you can place a loose vertex there and use the Shift+S menu then Cursor to Selected to snap it there.
Afterwards hide all geometry of your object by selecting it and pressing H (so it doesn't get unintentionally deformed), except for the beveled edges only.
Turn on Proportional Edit (O key by default), set Falloff Type to Linear then selecting only the tip vertex at 1 scale down as desired. Press Shift+X (Y or Z depending on the orientation of your model) to exclude the desired axis from scaling and prevent distortions, then adjust the influence radius with Mousewheel as desired.
In the end Alt+H to unhide hidden geometry
I'm not sure if you're using the Bevel modifier or function, but I would use the function inside of "Edit" mode. First of all, loop cut your cube into four like you have in your screenshot, and select all of the edges you want to bevel. Press space and type in "Snap Cursor to Active," and click on the result. Then, bevel the edges with the maximum weight you want applied (.5 for example). Then, set the pivot point to "3D Cursor" by using the drop down at the bottom of the screen. This is what it should look like at this step:
Next, select all of the edges you want to have the least bevel weight. Press S to scale, then Shift + X because you want the beveled edges to stay in the same relative X position. Type in the decimal percentage of the maximum weight you want these edges to have (.2 for example), and left click to apply the transformation. It looks a bit funky, but this is what this step should look like:
Keep on repeating the last step, choosing the next edges in line, and increasing the scaling percentage until you reach the maximum weight percentage (i.e. 1). Finally, your bevel should look like this:
In addition to the other great answers here, I'll share an approach I use that works well.
You can set key bevel points (I have four different bevels in this example) and then use Bridge Edge Loops to interpolate the geometry between them.
As you can see, depending on the settings you use you can create various types of curves.
If more complex curves are needed, a modifier such as Simple Deform or Curve can be used to further bend the geometry:
Bevel modifier provides Percent method of defining width which can fit this situation. Used together with bevel weight you can define different bevel width for different edges.
Select an edge and assign weight to it. It's better to go gradually from low to high or from high to low weights to keep track of them; it doesn't seem to be possible to do that with Proportional Edit. Add modifier to the object, set settings for Limit as Weight and Width as Percent as on the screenshot. Edge bevel below is shown on the mesh where blue means weight closer to 1, grey-black - closer to 0.
Bevel width with this method should be set to high values because it is used as percentage value.
Similar to Offset but the value is interpreted as a percentage of the adjacent edge length.