What actions can I take against my landlord?

The landlord would be liable, or responsible, for what occured only IF it was due to the landlord's negligence or fault, such as if the landlord knew of obstructions in one of the pipes under its control but did not correct that problem, and that's what led to the back-up. If the landlord was not at fault--for example, it occured because of a problem in the municipal part of the sewer line or system; or because another tenant (if there are any other tenants) flushed something he or she should not have--then the landlord is not liable or responsible.
If the landlord was at fault (e.g. you think it occured due to lack of maintenance, etc.) the landlord should pay for the clean-up. Unfortunately, there is no practical way to force them to do so or to recover the cost therefore other than a lawsuit, which takes weeks or months. You may need to pay for a commercial clean-up company to do the work, then look to sue the landlord, if the landlord will not reimburse you, to recover the cost.
If the landlord was not at fault, you can't recover any compensation from them. In theory, you could sue the city, if the city has been negligent; or a tenant who caused the problem if there was such and you can identify him or her. Again, though, those remedies can take time; you are probably left hiring a clean-up serice (if you are, understandably, not willing to do this yourself) and then suing later for the costs.