How can I create a self-signed certificate using C#?

This implementation uses the CX509CertificateRequestCertificate COM object (and friends - MSDN doc) from certenroll.dll to create a self signed certificate request and sign it.

The example below is pretty straight forward (if you ignore the bits of COM stuff that goes on here) and there are a few parts of the code that are really optional (such as EKU) which are none-the-less useful and easy to adapt to your use.

public static X509Certificate2 CreateSelfSignedCertificate(string subjectName)
{
    // create DN for subject and issuer
    var dn = new CX500DistinguishedName();
    dn.Encode("CN=" + subjectName, X500NameFlags.XCN_CERT_NAME_STR_NONE);

    // create a new private key for the certificate
    CX509PrivateKey privateKey = new CX509PrivateKey();
    privateKey.ProviderName = "Microsoft Base Cryptographic Provider v1.0";
    privateKey.MachineContext = true;
    privateKey.Length = 2048;
    privateKey.KeySpec = X509KeySpec.XCN_AT_SIGNATURE; // use is not limited
    privateKey.ExportPolicy = X509PrivateKeyExportFlags.XCN_NCRYPT_ALLOW_PLAINTEXT_EXPORT_FLAG;
    privateKey.Create();

    // Use the stronger SHA512 hashing algorithm
    var hashobj = new CObjectId();
    hashobj.InitializeFromAlgorithmName(ObjectIdGroupId.XCN_CRYPT_HASH_ALG_OID_GROUP_ID,
        ObjectIdPublicKeyFlags.XCN_CRYPT_OID_INFO_PUBKEY_ANY, 
        AlgorithmFlags.AlgorithmFlagsNone, "SHA512");

    // add extended key usage if you want - look at MSDN for a list of possible OIDs
    var oid = new CObjectId();
    oid.InitializeFromValue("1.3.6.1.5.5.7.3.1"); // SSL server
    var oidlist = new CObjectIds();
    oidlist.Add(oid);
    var eku = new CX509ExtensionEnhancedKeyUsage();
    eku.InitializeEncode(oidlist); 

    // Create the self signing request
    var cert = new CX509CertificateRequestCertificate();
    cert.InitializeFromPrivateKey(X509CertificateEnrollmentContext.ContextMachine, privateKey, "");
    cert.Subject = dn;
    cert.Issuer = dn; // the issuer and the subject are the same
    cert.NotBefore = DateTime.Now;
    // this cert expires immediately. Change to whatever makes sense for you
    cert.NotAfter = DateTime.Now; 
    cert.X509Extensions.Add((CX509Extension)eku); // add the EKU
    cert.HashAlgorithm = hashobj; // Specify the hashing algorithm
    cert.Encode(); // encode the certificate

    // Do the final enrollment process
    var enroll = new CX509Enrollment();
    enroll.InitializeFromRequest(cert); // load the certificate
    enroll.CertificateFriendlyName = subjectName; // Optional: add a friendly name
    string csr = enroll.CreateRequest(); // Output the request in base64
    // and install it back as the response
    enroll.InstallResponse(InstallResponseRestrictionFlags.AllowUntrustedCertificate,
        csr, EncodingType.XCN_CRYPT_STRING_BASE64, ""); // no password
    // output a base64 encoded PKCS#12 so we can import it back to the .Net security classes
    var base64encoded = enroll.CreatePFX("", // no password, this is for internal consumption
        PFXExportOptions.PFXExportChainWithRoot);

    // instantiate the target class with the PKCS#12 data (and the empty password)
    return new System.Security.Cryptography.X509Certificates.X509Certificate2(
        System.Convert.FromBase64String(base64encoded), "", 
        // mark the private key as exportable (this is usually what you want to do)
        System.Security.Cryptography.X509Certificates.X509KeyStorageFlags.Exportable
    );
}

The result can be added to a certificate store using X509Store or exported using the X509Certificate2 methods.

For a fully managed and not tied to Microsoft's platform, and if you're OK with Mono's licensing, then you can look at X509CertificateBuilder from Mono.Security. Mono.Security is standalone from Mono, in that it doesn't need the rest of Mono to run and can be used in any compliant .Net environment (e.g. Microsoft's implementation).

Since .NET 4.7.2 you can create self-signed certs using System.Security.Cryptography.X509Certificates.CertificateRequest.

For example:

using System;
using System.IO;
using System.Security.Cryptography;
using System.Security.Cryptography.X509Certificates;

public class CertificateUtil
{
    static void MakeCert()
    {
        var ecdsa = ECDsa.Create(); // generate asymmetric key pair
        var req = new CertificateRequest("cn=foobar", ecdsa, HashAlgorithmName.SHA256);
        var cert = req.CreateSelfSigned(DateTimeOffset.Now, DateTimeOffset.Now.AddYears(5));

        // Create PFX (PKCS #12) with private key
        File.WriteAllBytes("c:\\temp\\mycert.pfx", cert.Export(X509ContentType.Pfx, "P@55w0rd"));

        // Create Base 64 encoded CER (public key only)
        File.WriteAllText("c:\\temp\\mycert.cer",
            "-----BEGIN CERTIFICATE-----\r\n"
            + Convert.ToBase64String(cert.Export(X509ContentType.Cert), Base64FormattingOptions.InsertLineBreaks)
            + "\r\n-----END CERTIFICATE-----");
    }
}