I figured it out.

You have to first create a Marketing campaign through Tea Commerce, as far as I can see, you're not able to do this programatically using the Tea Commerce API.

When having created the Marketnig campaign, you have to add a Discount Code rule and an award, then you have to get the ID of the discount code rule. There's probably a way better way to get the ID, but the way I've done it so far is this:

    // TODO: REMOVE THIS IN PRODUCTION
    public ActionResult DiscountCodeGenerate(long storeId)
    {
        var allCampaigns = TeaCommerce.Api.Marketing.Services.CampaignService.Instance.GetAllActive(storeId);
        var campaign = allCampaigns.FirstOrDefault(i => i.Name == "NewsletterSignupDiscount");
        return Content(campaign.RulesGroups.FirstOrDefault().Rules.FirstOrDefault().Id.ToString());
    }

Next, you need to generate the actual code. So let's assume the rule ID is 1, you can execute the following code:

TeaCommerce.Api.Marketing.Services.DiscountCodeService.Instance.Generate(storeId, 1, 1, 1, 10, String.Empty, String.Empty);