Whether you're buying from a dealer or a private seller, it's always worth trying to negotiate on the price.
Whatever happens, stick to your maximum price. Try to keep as calm as you can and take your time when talking to the seller. Even if you really love the car, it's best not to let make this too obvious to the owner, as it won't put you in a very strong bargaining position. It's unlikely that they will accept your first offer, especially if it is considerably lower than the asking price. If they don't accept then ask them what their best price would be.
Try to base your negotiation on real faults that you have recorded during the inspection. List the things that would need work to get right and bear in mind whether road tax is included. Add up all of the faults and then knock them off of the price as accurately as you can. You can't reduce your offers once made, so creep up slowly if you need to.
If the seller does move on their original price, it's a sign that you've got them interested so don't just give in at the first hurdle. You should make another offer a little bit higher than your first (as long as it's still within your limit). If they say that they can't move any more you might have to try either involving a second person, such as your partner or compare the price with similar cars that you have seen.
When you make your final offer, you could see if the seller will split the difference and if you can pay cash, use that to your advantage. If they don't accept, remember that you need to stay in control and not be tempted to go over your maximum price. Leave your phone number so they can ring if they don't get a better offer