How can I Parse Json in a Azure Function

You answer above is returning a string and not JSON. I would suggest that you modify your answer as follows:

public static async Task<HttpResponseMessage> Run(HttpRequestMessage req, TraceWriter log)
{
    dynamic body = await req.Content.ReadAsStringAsync();
    var e = JsonConvert.DeserializeObject<EventData>(body as string);
    return req.CreateResponse(HttpStatusCode.OK, e);
}

This will return the Sample output without the JSON escapes:

{"Category":"Azure Functions","Action":"Run","Label":"Test"}

As for .Net Core 2 :

string requestBody = await new StreamReader(req.Body).ReadToEndAsync();

and then you can deserialize it:

dynamic jObject= JsonConvert.DeserializeObject(requestBody);

and to complete your answer (convert to object):

JToken jCategory = jObject;
var whatever = jCategory["yourkeyiteminjson"].ToObject<YourObjectType>();

i.e to show you how flexible it is:

Let's say for this Json input:

{"companyId": "123456","payments": [{"id": "31212"},{"id": "31212"},{"id": "3"}],"miFees": [{"id": "452"},{"id": "254"}]}

You can do as follow:

var companyId = jObject["companyId"].ToString();
var payments = jCategory["payments"].ToObject<List<PaymentTransaction>>();

Here is a complete Azure Function source code for serializing/deserializing objects using JsonNet:

#r "Newtonsoft.Json"

using System.Net;
using Newtonsoft.Json;

public static async Task<HttpResponseMessage> Run(HttpRequestMessage req, TraceWriter log)
{
    dynamic body = await req.Content.ReadAsStringAsync();
    var e = JsonConvert.DeserializeObject<EventData>(body as string);
    return req.CreateResponse(HttpStatusCode.OK, JsonConvert.SerializeObject(e));
}

public class EventData
{
    public string Category { get; set; }
    public string Action { get; set; }
    public string Label { get; set; }
}

Sample input (request body):

{
    "Category": "Azure Functions",
    "Action": "Run",
    "Label": "Test"
}

Sample output:

"{\"Category\":\"Azure Functions\",\"Action\":\"Run\",\"Label\":\"Test\"}"