How can I use 5 inputs in an sqlite query and have some of them empty?

First off, don't do it like that. Never concatenate strings to build a SQL command. It leaves you wide open to accidental or deliberate SQL Injection attack which can destroy your entire database. Always use Parameterized queries instead.

When you concatenate strings, you cause problems because SQL receives commands like:

SELECT * FROM MyTable WHERE StreetAddress = 'Baker's Wood'

The quote the user added terminates the string as far as SQL is concerned and you get problems. But it could be worse. If I come along and type this instead: "x';DROP TABLE MyTable;--" Then SQL receives a very different command:

SELECT * FROM MyTable WHERE StreetAddress = 'x';DROP TABLE MyTable;--'

Which SQL sees as three separate commands:

SELECT * FROM MyTable WHERE StreetAddress = 'x';

A perfectly valid SELECT

DROP TABLE MyTable;

A perfectly valid "delete the table" command

--'

And everything else is a comment.
So it does: selects any matching rows, deletes the table from the DB, and ignores anything else.

So ALWAYS use parameterized queries! Or be prepared to restore your DB from backup frequently. You do take backups regularly, don't you?

Then when you have fixed that through your whole app, build your SQL WHERE condition by using a List<string>: if the textbox is empty, do nothing.
If it isn't, add the appropriate column name and the LIKE info to the collection:

clauses.Add("Column5 LIKE '%@C5%');

Then add the column parameter to the SqLiteCommand.Parameters collection:

cmd.Parameters.AddWithValue("@C5", textboxPCR.Text);

After the textboxes are all dealt with, add the clauses to your SQL command:

cmd.CommandText = "SELECT * FROM PMinfo WHERE " + string.Join(" AND ", clauses);