It will depend on how you want to play the songs.
If you're simply playing them in iTunes, you can adjust the volume of the affected tracks. However, the last time I checked, that volume change does not affect the copy added to portable devices (iPod,iPhone etc.).
To adjust the volume for use in iTunes:
- select the song(s) and on iTunes' top bar menu select Edit/Song Info or use right-click the selection/Get Info. (You can select multiple songs)
- switch from the Details tab to the Options tab
- on the Options tab, use volume adjust to increase the volume of your selection. Note that it will not change the level of a song while it is playing. Instead, you would need to re-start the song
You might need to experiment a bit in order to find the correct level and it won't be perfect. The nature of the way music is recorded will cause one album to have a different perceived level than another. I.E. an album by band A, produced by producer B, may sound louder than an album by artist C, produced by producer D.
Be aware though, that this change has been made in iTunes, for use in iTunes. It does not affect the original file. So if (when) you use that file in another programme, it will sound quieter than your other music.
To change the level of the file, so that the change works in other music playing software, you would need to change the original file. This can be accomplished by a programme called Audacity (it's free), but be aware that it may take you a while to get used to it (see below*).
The Audacity programme can open most (but not all) formats of audio file. It can certainly open MP3 files, which is probably the format that your old LPs are in. Once in Audacity, it's possible to amplify the sound level (volume) and then save that change to the original location. The next time that iTunes plays the song, it will be at the new level. Getting that revised version onto a portable Apple device (such as an iPod) is a bit more involved, so think about that at a later time.
* For now, I'll explain Audacity so that you can see what is involved. The basic steps in Audacity that you would need to follow are:
- open the file for the song in Audacity
- the song is presented in "waveform" style. In other words, the music is shown as a waveform. Don't worry too much about what that means, just accept that the represents the music that you hear. See the screenshot:

- select the entire waveform that is shown
- amplify the waveform
- save the file
Obviously, there's a little more to it than that, such as the level to amplify it to, but we can discuss that later if you wish.