To unlock limit breaks in FFVII you need to score a certain number of kills, if I summon, do the kills the summon gets count towards this total?
Unlocking new limit breaks is different for each character, depending on how many limit breaks they have, how many levels they have, and how many of each pre-requisite needs to be met to unlock the next limit break, or the next level.
For example, everyone (except Cait Sith and Vincent) have a total of 7 limit breaks. Each Limit Level has two Limit Breaks. To unlock each Limit Break, the character either has to have killed a certain number of enemies, or used (a) limit break a certain amount of times.
Eg; to unlock the third Limit Level, Cloud either needs to have killed 320 enemies, or, used 7 limit breaks. Vincent, on the other hand, is not reliant on limit breaks, and only needs to have killed 40 enemies.
Then, once a character reaches their final Limit Level, a mini-quest to find their final Limit Break "Manual" appears.
Now, as to whether or not summons affect this - yes they do. They rack up on the kill count, therefore they will help unlock the next limit level.