How can I configure NLog in .NET Core with appsettings.json instead of an nlog.config file?

Yes, this is possible but has a minimum version requirement. You must be using NLog.Extensions.Logging >= 1.5.0. Note that for ASP.NET Core applications this will be installed as a dependency if you install NLog.Web.AspNetCore >= 4.8.2.

You can then create an NLog section in appsettings.json and load it with the following code:

var config = new ConfigurationBuilder()
  .SetBasePath(System.IO.Directory.GetCurrentDirectory())
  .AddJsonFile("appsettings.json", optional: true, reloadOnChange: true).Build();
NLog.Config.LoggingConfiguration nlogConfig = new NLogLoggingConfiguration(config.GetSection("NLog"));

For example, for an ASP.NET Core application, your Main() method in Program.cs should look something like this:

public static void Main(string[] args)
{
    var config = new ConfigurationBuilder()
        .SetBasePath(System.IO.Directory.GetCurrentDirectory())
        .AddJsonFile("appsettings.json", optional: true, reloadOnChange: true).Build();
    LogManager.Configuration = new NLogLoggingConfiguration(config.GetSection("NLog"));

    var logger = NLog.Web.NLogBuilder.ConfigureNLog(LogManager.Configuration).GetCurrentClassLogger();
    try
    {
        logger.Debug("Init main");
        CreateWebHostBuilder(args).Build().Run();
    }
    catch (Exception ex)
    {
        logger.Error(ex, "Stopped program because of exception");
    }
    finally {
        LogManager.Shutdown();
    }
}

A configuration like the one in the question can be achieved with the following settings in appsettings.json:

"NLog":{
    "internalLogLevel":"Info",
    "internalLogFile":"c:\\temp\\internal-nlog.txt",
    "extensions":{
        "NLog.Web.AspNetCore":{
            "assembly":"NLog.Web.AspNetCore"
        }
    },
    "targets":{
        "allfile":{
            "type":"File",
            "fileName":"c:\\temp\\nlog-all-${shortdate}.log",
            "layout":"${longdate}|${event-properties:item=EventId_Id}|${uppercase:${level}}|${logger}|${message} ${exception:format=tostring}"
        },
        "ownFile-web":{
            "type":"File",
            "fileName":"c:\\temp\\nlog-own-${shortdate}.log",
            "layout":"${longdate}|${event-properties:item=EventId_Id}|${uppercase:${level}}|${logger}|${message} ${exception:format=tostring}|url: ${aspnet-request-url}|action: ${aspnet-mvc-action}"
        }
    },
    "rules":[
        {
            "logger":"*",
            "minLevel":"Trace",
            "writeTo":"allfile"
        },
        {
            "logger":"Microsoft.*",
            "maxLevel":"Info",
            "final":"true"
        },
        {
            "logger":"*",
            "minLevel":"Trace",
            "writeTo":"ownFile-web"
        }
    ]
}

Edit: Thanks to Rolf Kristensen (who developed this functionality for NLog in the first place!) for pointing out this wiki page with more documentation on this feature: https://github.com/NLog/NLog.Extensions.Logging/wiki/Json-NLog-Config

Adding to Rolf Kristensen's response.

appsettings.json with database target.

"NLog": {
    "autoReload": true,
    "throwConfigExceptions": true,
    "internalLogLevel": "info",
     "internalLogFile": "c:\\temp\\internal-nlog.txt",
    "extensions": {
      "NLog.Extensions.Logging": {
        "assembly": "NLog.Extensions.Logging"
      }
    },

    "targets": {
      "database": {
        "type": "Database",
        "commandText": "INSERT INTO dbo.log (MachineName,Logged,Level,Message,Logger,Callsite,Exception) values (@MachineName,@Logged,@Level,@Message,@Logger,@Callsite,@Exception)",

        "parameters": [

          {
            "name": "@MachineName",
            "layout": "${machinename}"
          },
          {
            "name": "@Logged",
            "layout": "${date}"
          },
          {
            "name": "@Level",
            "layout": "${level}"
          },
          {
            "name": "@Message",
            "layout": "${message}"
          },
          {
            "name": "@Logger",
            "layout": "${logger}"
          },
          {
            "name": "@Callsite",
            "layout": "${callsite}"
          },
          {
            "name": "@Exception",
            "layout": "${exception:tostring}"
          }
        ],
        "dbProvider": "System.Data.SqlClient",
        "connectionString": "Data Source=database server;Initial Catalog=database ;Trusted_Connection=False; User Id=AppUser;Password=AppUserPassword;"
      }
    },

    "rules": [
      {
        "logger": "*",
        "minLevel": "Trace",
        "writeTo": "database"
      }
    ]
  }