ASP.NET 5 (aka ASP.NET Core), MVC Core Solution
In ASP.NET Core, things are further optimized as we don't need to stringify the markup in the extension method.
In ~/Extesions/HtmlExtensions.cs:
using System.Text.RegularExpressions;
using Microsoft.AspNetCore.Html;
using Microsoft.AspNetCore.Mvc.Rendering;
namespace YourProjectNamespace.Extensions
{
    public static class HtmlExtensions
    {
        private static readonly HtmlContentBuilder _emptyBuilder = new HtmlContentBuilder();
        public static IHtmlContent BuildBreadcrumbNavigation(this IHtmlHelper helper)
        {
            if (helper.ViewContext.RouteData.Values["controller"].ToString() == "Home" ||
                helper.ViewContext.RouteData.Values["controller"].ToString() == "Account")
            {
                return _emptyBuilder;
            }
            string controllerName = helper.ViewContext.RouteData.Values["controller"].ToString();
            string actionName = helper.ViewContext.RouteData.Values["action"].ToString();
            var breadcrumb = new HtmlContentBuilder()
                                .AppendHtml("<ol class='breadcrumb'><li>")
                                .AppendHtml(helper.ActionLink("Home", "Index", "Home"))
                                .AppendHtml("</li><li>")
                                .AppendHtml(helper.ActionLink(controllerName.Titleize(),
                                                          "Index", controllerName))
                                .AppendHtml("</li>");
            if (helper.ViewContext.RouteData.Values["action"].ToString() != "Index")
            {
                breadcrumb.AppendHtml("<li>")
                          .AppendHtml(helper.ActionLink(actionName.Titleize(), actionName, controllerName))
                          .AppendHtml("</li>");
            }
            return breadcrumb.AppendHtml("</ol>");
        }
    }
}
~/Extensions/StringExtensions.cs remains the same as below (scroll down to see the MVC5 version).
In razor view, we don't need Html.Raw, as Razor takes care of escaping when dealing with IHtmlContent:
....
....
<div class="container body-content">
    <!-- #region Breadcrumb -->
    @Html.BuildBreadcrumbNavigation()
    <!-- #endregion -->
    @RenderBody()
    <hr />
...
...
ASP.NET 4, MVC 5 Solution
=== ORIGINAL / OLD ANSWER BELOW ===
(Expanding on Sean Haddy's answer above)
If you want to make it extension-driven (keeping Views clean), you can do something like:
In ~/Extesions/HtmlExtensions.cs:
(compatible with MVC5 / bootstrap)
using System.Text;
using System.Web.Mvc;
using System.Web.Mvc.Html;
namespace YourProjectNamespace.Extensions
{
    public static class HtmlExtensions
    {
        public static string BuildBreadcrumbNavigation(this HtmlHelper helper)
        {
            // optional condition: I didn't wanted it to show on home and account controller
            if (helper.ViewContext.RouteData.Values["controller"].ToString() == "Home" ||
                helper.ViewContext.RouteData.Values["controller"].ToString() == "Account")
            {
                return string.Empty;
            }
            StringBuilder breadcrumb = new StringBuilder("<ol class='breadcrumb'><li>").Append(helper.ActionLink("Home", "Index", "Home").ToHtmlString()).Append("</li>");
            breadcrumb.Append("<li>");
            breadcrumb.Append(helper.ActionLink(helper.ViewContext.RouteData.Values["controller"].ToString().Titleize(),
                                               "Index",
                                               helper.ViewContext.RouteData.Values["controller"].ToString()));
            breadcrumb.Append("</li>");
            if (helper.ViewContext.RouteData.Values["action"].ToString() != "Index")
            {
                breadcrumb.Append("<li>");
                breadcrumb.Append(helper.ActionLink(helper.ViewContext.RouteData.Values["action"].ToString().Titleize(),
                                                    helper.ViewContext.RouteData.Values["action"].ToString(),
                                                    helper.ViewContext.RouteData.Values["controller"].ToString()));
                breadcrumb.Append("</li>");
            }
            return breadcrumb.Append("</ol>").ToString();
        }
    }
}
In ~/Extensions/StringExtensions.cs:
using System.Globalization;
using System.Text.RegularExpressions;
namespace YourProjectNamespace.Extensions
{
    public static class StringExtensions
    {
        public static string Titleize(this string text)
        {
            return CultureInfo.CurrentCulture.TextInfo.ToTitleCase(text).ToSentenceCase();
        }
        public static string ToSentenceCase(this string str)
        {
            return Regex.Replace(str, "[a-z][A-Z]", m => m.Value[0] + " " + char.ToLower(m.Value[1]));
        }
    }
}
Then use it like (in _Layout.cshtml for example):
....
....
<div class="container body-content">
    <!-- #region Breadcrumb -->
    @Html.Raw(Html.BuildBreadcrumbNavigation())
    <!-- #endregion -->
    @RenderBody()
    <hr />
...
...