Skin id goes in:
"New Icon (Skin)": SKINIDNUMBERHERE

Take for example this skin:
https://steamcommunity.com/sharedfiles/filedetails/?id=2124528682
The skins id # is in the links, here is the id # of that skin above: 2124528682

Custom display name can be whatever you want it, and goes here for name:
"Display Name": "Any Name You Want",

{
  "Values": [
    {
      "Shortname": "scientistsuit_heavy",
      "Display Name": "!!! HEAVY SUIT !!!",
      "New Icon (Skin)": 0
    },
    {
      "Shortname": "rifle.ak",
      "Display Name": "TOP GUN, ORANGE#1",
      "New Icon (Skin)": 0
    }
  ]
}

1801450909

After testing, multiple names with skin ID cannot be displayed

What item are you using and which skin for it?
I'll check on my end as well

Ah yes, currently supports only labeling 1 item with skin at a time.
Tested mine here:

{
"Values": [
{
"Shortname": "grenade.f1",
"Display Name": "Neon F1 Frag Grenande",
"New Icon (Skin)": 1117908964
},
{
"Shortname": "rifle.ak",
"Display Name": "Badass AK47",
"New Icon (Skin)": 2207842553
}
]
}

works, just cannot multi label items.

If you can have multiple display names, that's great. The plug-in has unlimited potential

Because that plugin is for general replacement of item