Like others have said, we will need somewhere to start on how you play Lucina to help you improve. But basic advice that I can give you for any character in general is to go to training mode and practice hitting the sand bag, and ask what you want from the character. Practice the spacing of their moves and look at the lag of each individual attack and what would work best in each situation.
Lucina is a character that plays in what I have heard refer to "aggressive passive" as opposed to "passive aggressive" where you want to bait out openings to string together some attacks and then once they are done, you want to reset to neutral and go back to baiting out the opponent to make a mistake and give you an opening. In my opinion, Fairs and tilts are probably going to be your biggest ally in this, and utilizing different timings on dancing blade is a really good way to catch your opponent off guard and rack up some damage. Down throw is also a really good way to string together attacks at lower percentages. Up B out of shield is also a great tool that a lot of people don't expect. It is almost instant, and as long as you get the spacing down, you aren't going to be punished for it most of the time because it has a decent launch distance. Last but not least, try to fake them out with a shield breaker every once and awhile. If you condition them to shield your attacks, they are either going to end up getting their shield broken which is a near guaranteed kill above around 60% depending on your stage position, or take heavy damage from it because they drop their shield or don't dodge in time.
From there on out, it is all about getting either a good read with a smash attack/countering a strong hit, or utilizing her range in the air to gimp the opponent if you can get them of stage. Keep in mind, that while she does have a spike, unless you can get a really good read on where they are going to be, then an FAir or BAir is probably going to be more beneficial to you off stage than her DAir is because it is one of the harder spikes to land in the game, and if you miss the sweet spot at the bottom of her feet, they aren't going to be going as far away from the ledge as they would have if you hadn't tried to be flashy. The DAir also has a considerable amount of lag on it, so using it off stage can be incredibly dangerous as well.
That is really Lucina in a nut shell. There are good guides on the site that go into more detail, I think Eternal linked you his, so if you have time you should go ahead and read that, as well as watch some of the videos in the Lucina video compendium.