How do I waterproof a foundation from the inside?

I think you already know the biggest source of the problem, the ground should slope away from the house with at least a 5% grade. That will divert water away from the foundation; it seems right now you may be directing it towards your foundation, that will exert hydro-static pressure on your foundation walls, and possibly come in through a failure in your exterior membrane. See the following page has diagrams on how to build basements properly - 

How to insulate a basement and protect it against water damage and flooding

Bulk water directed towards your foundation will also raise the water table, so it's not a great surprise that it will find its way up through the junction between walls and the floor. It’s hard to give you accurate advice for interior repairs without more details about what insulation and finishing you have, but proper exterior drainage and other storm water protection may solve the problem so definitely start with that and see if it solves the problem. 

Some of these features you may already have in place, but here is a short list anyway:

• Slope the ground away from the foundation
• Direct storm water towards a drainage ditch or a drywell with swales or drains as needed.
• Install eaves troughs, being sure to divert water far from the house

Those are the outside solutions we would recommend, as for interior, if you don’t already have one, consider installing a sump pump to keep the water table below the level of the foundation floor. If these solutions don’t solve the problem, please write back and include details about your basement walls and floors – insulation, vapour barrier, finishing materials, how it is assembled, etc. If your basement currently has no finished flooring, an interior water-proofing solution is much easier, it's just that without knowing its current state it’s hard for us to offer accurate advice as to how else to move forward, though hopefully you may not need to.