1. 确保Serilog、Serilog.Sinks.Graylog、Serilog.Exceptions、Serilog.Settings.Configuration等NuGet包已经安装。
2. 在appsettings.json文件中配置Serilog以及Graylog地址、端口等信息。
3. 在程序启动时对Serilog进行配置，并添加Graylog作为日志输出源。
4. 在代码中如果需要记录异常信息，则需要在相应的Catch块中调用Serilog对应的异常记录方法。

代码示例：

1. appsettings.json配置

"Serilog": {
    "Using": ["Serilog.Sinks.Graylog"],
    "MinimumLevel": {
        "Default": "Information",
        "Override": {
            "Microsoft": "Warning",
            "System": "Warning"
        }
    },
    "WriteTo": [
        {
            "Name": "Graylog",
            "Args": {
                "facility": "AspNetCoreSample",
                "serverUrl": "http://localhost:12201",
                "transportType": "Udp",
                "minimumLogEventLevel": "Information",
                "maximumMessageSize": 8192,
                "stackTraceDepth": 10,
                "batchSizeLimit": 100,
                "period": "00:00:01",
                "includeMessageTemplate": true,
                "messageIdGenerator": "Serilog",
                "messageGenerator": "Serilog.Sinks.Graylog.Core.MessageBuilders.PropertyMessageGenerator, Serilog.Sinks.Graylog.Core",
                "formatProvider": "Serilog.Sinks.Graylog.Core.FormatProviders.StructurelessJsonFormatter, Serilog.Sinks.Graylog.Core"
            }
        }],
    "Enrich": ["WithExceptionDetails"]
}

2. 程序启动时对Serilog进行配置

public static IHostBuilder CreateHostBuilder(string[] args) =>
    Host.CreateDefaultBuilder(args)
        .UseSerilog((hostingContext, loggerConfiguration) =>
        {
            loggerConfiguration.ReadFrom.Configuration(hostingContext.Configuration)
                .Enrich.FromLogContext();
        })
        .ConfigureWebHostDefaults(webBuilder =>
        {
            webBuilder.UseStartup();
        });

3. 记录异常信息

try 
{
    // Something may cause exceptions
} 
catch (Exception ex) 
{
    _logger.LogError(ex, "An error has occurred"); //记录异常日志
}