I thought it was rather clear what they were trying to do - River was one of several children who were identified as having potential psychic abilities. They were experimented on to enhance and develop those abilities.
Their apparent purpose was to create a number of telepathic spies and/or secret agents. These people could be used in any number of ways, ranging from internal security to infiltration and even assassination. A telepath would excel at any of those - they could detect anyone in your organization who was planted as a spy (or who had developed disloyalty), could easily infiltrate any organization (it's easy to get hired if you know exactly what the interviewer wants to hear), and assassination would be simple for someone who knows exactly how to get through your security and how to become your best friend.
The changes done to River's mind (stripping the amygdala, among other things) were part of this experimentation process. In the series, we learned that this process had stripped away her ability to push aside her emotions.
This is obviously not a desirable trait in a psychic spy, so presumably the Alliance thought they could reverse the damage once they'd enhanced her abilities. That was my initial thought, at least, until I saw Serenity.
When River is exposed to the commercial that triggers her, she seems to go almost completely emotionless. Perhaps the Alliance found a way to actually suppress her emotions - if she couldn't feel any emotion they didn't let her feel, she would have become the perfect agent. She would have been completely loyal, utterly merciless, and without regard for her own safety.
River wasn't turned into an assassin, she was simply turned into the best Operative+ they could make, and her psychic abilities were enhanced. The fact that this made her a brilliant potential assassin and an unstoppable hand-to-hand combatant was merely icing on the cake.
As for being a perfect weapon against Reavers, I strongly doubt that was the Alliance's intentions. To the Alliance as a whole, the Reavers were almost certainly considered a manageable threat.
Consider the Reaver's tactics: They travel in small groups (usually single ships closer in size to Serenity than to Alliance Cruisers). They raid planets and moons, using shock tactics and fury. They charge directly into hand-to-hand combat, eschewing most projectile weapons. They have a certain amount of cunning, but lack any overall strategic guidance or overt command structure. They did, absolutely have larger ships in their fleet, but it's notable that we never see any sign of this until people move towards Miranda.
It's probable that the Alliance thought they could handle the Reaver threat quite easily with their fleet and existing ground forces, so I doubt they would send such valuable assets as River against them intentionally.
A disciplined, prepared military would rip them to shreds.