During the kinescope era (until 1976) there were 14 league championship series played. In the National League, that accounts for 27 games. Not a single full broadcast copy of any of those 27 games exists. In fact, the only American League Championship Series broadcast that exists prior to 1976 is of Game 2 of the 1972 series.
What we have left is snippets, brief little glimpses of actual broadcast footage. Most of these continue to exist thanks to whatever footage was used for nightly news broadcasts. Generally, those feature only the audio from the actual sports anchor in the studio and not the original television crew.
So, where to begin? The first LCS seems as good a place as any. 1969 is a year that features the oldest known copy of a color World Series broadcast. Luckily, some enterprising people had the warewithal to preserve the color 'truck feed' tapes from the 3 World Series games played at Shea Stadium. The first two games in Baltimore are preserved via CBC kinescopes. Due to this, we are able to see this series in it's entirity.
Yet, those same enterprising folks did not seem to see fit to preserve the first ever League Championship Series. The Miracle Mets were pitted against the West Champion Atlanta Braves and their offense with Hammerin' Hank and The Baby Bull Cepeda in the middle of the lineup. The Mets made quick work of said Braves in 3 games.
It is that Game 3, at Shea Stadium again, that we have SOME color footage. What we have is a Hank Aaron HR in the 1st inning off of Gary Gentry:
It was a 2 run homerun as you can see Tony Gonzalez waiting at home along with the following batter Rico Carty.
The only other broadcast glimpse we get is of Braves starter Pat Jarvis. The clip is spliced together with some MLB Films footage in the Baseball Seasons: 1969 broadcast that all these shots come from. In that scene, Mets 2B Ken Boswell takes Jarvis deep. This happened in the bottom of the 4th inning.