How to deserialize an array of values with a fixed schema to a strongly typed data class?

The converter from Deserializing JSON in Visual Basic .NET should do what you need, suitably translated from VB.NET to c#:

public class ObjectToArrayConverter<T> : JsonConverter
{
    public override bool CanConvert(Type objectType)
    {
        return typeof(T) == objectType;
    }

    public override void WriteJson(JsonWriter writer, object value, JsonSerializer serializer)
    {
        var objectType = value.GetType();
        var contract = serializer.ContractResolver.ResolveContract(objectType) as JsonObjectContract;
        if (contract == null)
            throw new JsonSerializationException(string.Format("invalid type {0}.", objectType.FullName));
        writer.WriteStartArray();
        foreach (var property in SerializableProperties(contract))
        {
            var propertyValue = property.ValueProvider.GetValue(value);
            if (property.Converter != null && property.Converter.CanWrite)
                property.Converter.WriteJson(writer, propertyValue, serializer);
            else
                serializer.Serialize(writer, propertyValue);
        }
        writer.WriteEndArray();
    }

    public override object ReadJson(JsonReader reader, Type objectType, object existingValue, JsonSerializer serializer)
    {
        var contract = serializer.ContractResolver.ResolveContract(objectType) as JsonObjectContract;
        if (contract == null)
            throw new JsonSerializationException(string.Format("invalid type {0}.", objectType.FullName));

        if (reader.MoveToContentAndAssert().TokenType == JsonToken.Null)
            return null;
        if (reader.TokenType != JsonToken.StartArray)
            throw new JsonSerializationException(string.Format("token {0} was not JsonToken.StartArray", reader.TokenType));

        // Not implemented: JsonObjectContract.CreatorParameters, serialization callbacks, 
        existingValue = existingValue ?? contract.DefaultCreator();

        using (var enumerator = SerializableProperties(contract).GetEnumerator())
        {
            while (true)
            {
                switch (reader.ReadToContentAndAssert().TokenType)
                {
                    case JsonToken.EndArray:
                        return existingValue;

                    default:
                        if (!enumerator.MoveNext())
                        {
                            reader.Skip();
                            break;
                        }
                        var property = enumerator.Current;
                        object propertyValue;
                        // TODO:
                        // https://www.newtonsoft.com/json/help/html/Properties_T_Newtonsoft_Json_Serialization_JsonProperty.htm
                        // JsonProperty.ItemConverter, ItemIsReference, ItemReferenceLoopHandling, ItemTypeNameHandling, DefaultValue, DefaultValueHandling, ReferenceLoopHandling, Required, TypeNameHandling, ...
                        if (property.Converter != null && property.Converter.CanRead)
                            propertyValue = property.Converter.ReadJson(reader, property.PropertyType, property.ValueProvider.GetValue(existingValue), serializer);
                        else
                            propertyValue = serializer.Deserialize(reader, property.PropertyType);
                        property.ValueProvider.SetValue(existingValue, propertyValue);
                        break;
                }
            }
        }
    }

    static IEnumerable<JsonProperty> SerializableProperties(JsonObjectContract contract)
    {
        return contract.Properties.Where(p => !p.Ignored && p.Readable && p.Writable);
    }
}

public static partial class JsonExtensions
{
    public static JsonReader ReadToContentAndAssert(this JsonReader reader)
    {
        return reader.ReadAndAssert().MoveToContentAndAssert();
    }

    public static JsonReader MoveToContentAndAssert(this JsonReader reader)
    {
        if (reader == null)
            throw new ArgumentNullException();
        if (reader.TokenType == JsonToken.None)       // Skip past beginning of stream.
            reader.ReadAndAssert();
        while (reader.TokenType == JsonToken.Comment) // Skip past comments.
            reader.ReadAndAssert();
        return reader;
    }

    public static JsonReader ReadAndAssert(this JsonReader reader)
    {
        if (reader == null)
            throw new ArgumentNullException();
        if (!reader.Read())
            throw new JsonReaderException("Unexpected end of JSON stream.");
        return reader;
    }
}

Next, add the converter to your Player class, and indicate the order of each property using JsonPropertyAttribute.Order:

[JsonConverter(typeof(ObjectToArrayConverter<Player>))]
public class Player
{
    [JsonProperty(Order = 1)]
    public int UniqueID { get; set; }
    [JsonProperty(Order = 2)]
    public string PlayerDescription { get; set; }
    // Other fields as required.
}

Then finally, declare your root object as follows:

public class ScoreboardResults
{
    public int timestamp { get; set; }
    public int total_players { get; set; }
    public int max_score { get; set; }
    public Dictionary<string, Player> players { get; set; }
}

Note that I have moved Username out of the Player class and into the dictionary, as a key.

Note that data contract attributes can be used instead of Newtonsoft attributes to specify order:

[JsonConverter(typeof(ObjectToArrayConverter<Player>))]
[DataContract]
public class Player
{
    [DataMember(Order = 1)]
    public int UniqueID { get; set; }
    [DataMember(Order = 2)]
    public string PlayerDescription { get; set; }
    // Other fields as required.
}

Demo fiddles here, here and here.