Some large, export-oriented Danish companies, like Novo Nordisk and Lundbeck, have English as their corporate language and are happy to employ qualified non-Danes. They are great companies to work for, with great benefits, and therefore receive a lot of applications from both inside and outside Denmark.
Working off-the-books in a restaurant or as a cleaning person is possible, although illegal and not always reliable.
If you are coming to Denmark to be with a boyfriend or girlfriend, getting married will not necessarily make things easier for you, work-wise. In fact, your partner will have to prove to the Danish government that he or she can support you before you can get a "fiancee visa" - and since you won't be eligible for government support, you should plan on living off your partner for at least a year or two before you know enough Danish to get a full-time job. Can your relationship survive that?