What were the weaknesses in Lamark's theory of evolution?

His theory meant an animal that acquired a trait in it's lifetime, rather than through it's genetic makeup when conceived, could be passed on.

An acquired trait, for example, would be if a naturally scrawny guy worked out a lot and became very muscular. Lamarck's idea of evolution is that since he used his muscles so much, his offspring would be born with those traits, in this case a muscular child, rather than a scrawny one.

The problem with this is acquired traits aren't what is passed on to an organism's offspring, genetic code is, and that is determined at conception, not during one's lifetime.

To be fair, he was around before Darwin and Mendel and the discovery of DNA, and it wasn't a bad idea in my opinion. It just turned out not to be true.

There was nothing intrinsically wrong with his theory - it just turned out to be wrong. But it was a perfectly good theory given the available evidence he had at the time. That's how science works. You come up with a theory that explains what you know to be true. Then that theory makes predictions that you can test with experiments. In this case, you could take, for instance, two dogs. Work one really hard and let the other be lazy its whole life, and then measure their offspring. If the theory is right, the hard working dog will have muscular offspring, and the lazy dog will have weak offspring. You do the experiment and find that the prediction doesn't hold true, so your theory is wrong. And you come up with a better theory to explain your new set of facts.

As an aside, to claim that epigenetics shows that Lamarck was right is a HUGE overstatement of the truth. Epigenetics is simply a matter of gene regulation. It affects turning genes on and off. It does NOT change the genetic makeup itself, which is what Lamarckism would predict.

As far as I can see none.

People straw maned his position saying that if a cat looses its tail the offspring would be born without tails. Lamark was a one of the best biologists of his time and he certainly didn't think that.

We now know that he was right. Epigenetics is precisely that.