If you are using weight layers from the alpha trees UI, see Distribution
Navigate to the weight layers panel in the N-panel
Add a new slot by pressing the "+" button next to the layer stack list.
Select an output by going to the "Output" layer below and either selecting a vertex group from the dropdown or adding a new one with the "Add vertex group" button.
Add a new layer by clicking the "Add layer" button, and selecting one from the dropdown. Here I chose the Procedural texture layer.
You can now change all the parameters of the layer to your liking.
If you want to paint on the vertex group, add an "Input" layer, and select an input vertex group.
Tips and tricks
If you set the blend type of a layer to multiply, it will act as a mask, where zero values will all be zero, and one values will show through fully.
Adding a colour ramp adjustment is a good way to easily edit the contrast, range and brightness of a layer.
If weight layers is very slow because of a high poly mesh, you can enable manual mode, which lets you edit settings quickly, and only refresh the vertex group when you need to
If you add a layer stack from Alpha Trees, some buttons will be greyed out, as they are controlled by Alpha Trees.
If you want to edit the vertex group destructively, you can bake the layer stack, which is similar to applying a modifier, and set's all the vertex groups values permanently
Layer order: Whether to show the layers with the output at the bottom like in the modifiers or with the output at the top, like in Photoshop or Affinity photo layers.
Panel location: Whether to place the panel in the N-panel of the 3D view, or in the mesh data section of the properties panel (Next to the vertex groups)
Panel category: The category to place the panel in the N-panel