How can I fix my kitchen sink cheaply?
Since you already have a suspicion about the cause, the first step would be to confirm that.  The drain should slope at a rate of 1/4" per foot.  So if you had a 4' drain, there should be a difference of 1" in height between the start and end.  If its negative then that's without a doubt the issue. If this is the cause, the only way to fix it is to re-plumb it.  If you are handy, it is certainly doable.

If the slope isn't the issue then you have to start looking elsewhere.  It is probably worth confirming that the trap is clear of obstructions. Remove the trap below the sink and clean it out.  

If that fails then while it could still be a clog, its more likely a venting issue as BMitch suggested.

This isn't likely to be a big job for a plumber, so it might be most cost effective just to have one come out and do it right versus buying all the tools and parts needed.