"I wonder if it is me that is making her so unhappy"
Please don't think that. To your baby you are the source of all comfort and security. When they are little, their mums (or other primary caregiver) are the centre of their world.
As other posters have said, there can be lots of things that might upset your LO. I found it helpful to have a checklist in my head: hungry? thirsty? too warm? too cold? tired/overstimulated? bored? soiled nappy? etc.)
FWIW I don't think a 5mth old baby is going to be "bored" (assuming she is not left in her cot all day!) - if you take her from room to room with you, have interesting things to touch and look at, take her out in a sling or her buggy/pram for a change of scenery, put some nice music on, sing to her, talk to her, do action rhymes, etc. There is only so much a little baby can take on too, so don't feel you have to be singing & dancing all day long. Babies and toddlers need time to relax and rest, stretch out and experiment with the noises and movements they can make - psychologists think they consolidate their learning during these quiet times.
But please, please, please, don;t think you are to "blame" - if you feel negative about your mothering skills, take heart that all new mums find it terrifying and worrying at times, and we all have moments where we agonise over simple decisions and worry about doing the wrong thing. But as the months pass you will gain confidence in your new role, and your little girl will increasingly be able to communicate her needs and wishes to you - and life will be a bit easier.