Tear down that fence! How dare you want to put a barrier to prevent people from getting in...
Oh wait, wrong forum.
All kidding aside, are you going to do this yourself? If so, and you don't mind the look of vinyl, that is pretty easy (and fairly cheap) to install, and on a lot that size, it should be doable in a day.
If the vinyl look bothers you, then you can go for composite (getting more expensive), some of those look like real wood.
I wouldn't go with the wood route, way too much of a PITA to maintain.
Chain link is still good, it is pretty strong as well. Downside of this material is that it will eventually rust out.
If you want a rough idea on the costs, homedepot and lowes offer free estimates.
As for plants, I would say put gravel or <whatever> chips around the plants, this is so you don't have to take a trimmer to the fence all the time. A farmer told me that they use mint plants to keep the rodents away. Dwarf trees / shrubs could also be good, depends on your taste, and how much upkeep you want to do. 1 plant/shrub/bush in the middle of a vinyl fence is plenty.
The thing about trees that big is, they will grow. And then they could grow over the fence... then you get bitching from the neighbor.
Oh, and are you *sure* that fence is yours? Yes, the realtor might have said it was, and it might have been in the paperwork, but, unless you get a survey, you might end up having to tear the new fence out if your neighbor gets a survey, and the fence is on his side.