I am not sure if a HC will entertain such an application unless you have exhausted alternate remedies... Perhaps you can try couple of things before approaching the court..
File an RTI:
1. Get current status on your FIR and few FIRs before yours and few FIRs after your FIR.
2. Get copies of the case diaries for each of the above FIRs (all)
3. Get copies of all 161 statements for each of the above FIRs.
The above will give you ample evidence to show that the police is intentionally not investigating or completing it on time as envisaged in S.173 of CrPC.
With this you can file a petition with the higher police authorities and send them few reminders... after few months then you can approach the court.
Whether you can approach trial court for the same is not clear to me... but surely you can approach the High Court after you have exhausted the above remedies...
Hope this helps!