Q: How honest should I be with my current employer regarding my job search? 


Most people conduct a confidential search, which means you don’t let your employer know until you provide notice that you’re leaving. Any other choice would be a personal decision, based on your relationships atyour current job.
It could be if you let them know you feel that you have to look at the market, they’ll give you a raise to stay. It could be that they’d ask you to leave immediately.  
If there’s someone you feel you can take into your confidence where you are, because you feel personally obligated to provide a heads up (and perhaps, then, network through that person’s suggestions and with their colleagues outside the firm), you can choose to do that. However, in that anyone’s search can take some time, that approach could also carry some risk.  
Use your best judgment, with all that in mind. I hope it helps.