I'm sorry to hear of the situation. You absolutely should let your GP know that your depression is worsening and should also request a consultation with a Psychiatrist. GPs only have general knowledge of mental illness and a GP really shouldn't be supervising your treatment. While the emergence of suicidal thought is an indication that Depression is becoming quite serious it does not necessarily mean that hospitalization is required. Hospitalization is required when a person is believed to be at serious risk of harm . Each clinician will perform their own risk assessment but as a general statement assessment of suicide risk is not solely dependent upon suicidal ideation. Suicide risk assessment also considers whether a person has the means to act, has formed a plan and whether they have made a conscious decision to act. Many people experience suicidal ideation but have no intention to act, or don't have the means or haven't made a plan.
If you are concerned about how your GP may react then it would be reasonable for you to ask him/her about how they do a risk assessment before you disclose any information about suicidal ideation. I've had many discussions with patients who (like you) wish to seek help but are concerned about what will happen if they disclose suicidal ideation. In these circumstance I explain to them how I do the assessment and at what point I would request hospitalization. It is then up to them as to whether they will disclose information (they nearly always do as they realize that they are not going to be admitted).
I hope this has been of some help. Please let me know if you have further questions or would like me to clarify any part of my answer.
The feelings of hopelessness and thinking you are a burden are actually classic symptoms of Depression. I certainly think that you should consulting with someone who specializes in mental health problems, a GP is not usually a person who specializes in mental health. It sounds like your Depression is more than mild and you should certainly be seeing a specialist at this time. The recommended treatment for moderate Depression is an anti-depressant AND Cognitive Behavioral Therapy (CBT). The anti-depressant should be prescribed and supervised by a Psychiatrist and your GP should arrange a referral to one for you.
CBT is usually provided by Clinical Psychologists (although not exclusively) and you can contact the American Psychology Association (APA) for assistance with locating a Psychologist; take a look at the APA locator service here. You can use this to find Psychologists in your area and there is a phone number you can contact if you want a referral arranged for you. Also, take a look at an article published by the APA here. It's an interview with a senior Psychologist and covers some of the things you should consider when you looking for a Psychologist. The key point here is that you consult with a specialist ASAP!