Your landlord has an obligation to maintain the premises in a reasonably safe condition. You will have to prove that your landlord either (i) knew of the existence of the dangerous condition or (ii) should have known of the dangerous condition, and did not take reasonable steps to prevent it.
You will also need to prove that your injuries were caused as a result of your fall. Keep in mind that just because your landlord may be liability under the law, you still have to (i) prove your case, (ii) have sufficient injuries/damages to make filing suit worth your time, and be reasonably certain that there is insurance or equity in the real estate with which to satisfy any eventual judgment.
Do not hesitate to contact my office if you would like to discuss this matter further.
In order to recover in a lawsuit against your landlord, you will have to prove that the landlord either created the dangerous condition, knew of his existence, or reasonably should have known. Given the limited facts you provided, the lighting may be your stronger argument as opposed to be trash being left on the stairs by another tenant, unless the trash was there long enough that the landlord was aware of it or reasonably should have been based on reasonably periodic inspections. Another issue to consider is whether there were other defects to the stairs which contributed to your fall such as a loose hand rail.
A roundup of the best tips and legal advice.