Legacy Data and Current System Data Merged together in a single SQL Database
Obviously table1 is not a very good name, so I'll leave that as that. The other thing that immediately jumps at me is the poor formatting. 

It's hard to read multiple items strung across a single line. Doubly when they're aliased.

SELECT 
    CaseParticipantTable.DOCK_NO AS CaseNbr, ActivityTable.JUDGMENT_DATE AS JudgmentDate, ActivityTable.AMOUNT_TYPE AS AmountType, 
        ActivityTable.AMOUNT AS JudgmentAmount, CaseParticipantTable.FILING_NAME AS CaseParticipant


Is easier to read like this:

SELECT 
    CaseParticipantTable.DOCK_NO AS CaseNbr, 
    ActivityTable.JUDGMENT_DATE AS JudgmentDate, 
    ActivityTable.AMOUNT_TYPE AS AmountType, 
    ActivityTable.AMOUNT AS JudgmentAmount, 
    CaseParticipantTable.FILING_NAME AS CaseParticipant


Similarly, I don't like having the entire JOIN...ON statement on one line. I usually put the ON statement on the next line, indented one tab.

FROM
    UJSJudgementQuery.dbo.CJ_SC_CASE_PARTICIPANT AS CaseParticipantTable
    INNER JOIN UJSJudgementQuery.dbo.CJ_SC_JUDGMENT_ACTIVITY AS ActivityTable 
        ON CaseParticipantTable.DOCK_NO = ActivityTable.DOCK_NO


I do like how you start an indented new line after keywords, but you don't do it consistently. This WHERE suffers from that. I also recommend a new line for the items in your in List.

WHERE 
    CaseParticipantTable.FILING_NAME IN (
                                    '/CountyName COUNT AUDITORS OFFICE', 
                                    '/CountyName COUNTY AUDITOR', 
                                    '/CountyName COUNTY AUDITOR''S OFFI', 
                                    '/CountyName COUNTY AUDITORS OFFI', 
                                    '/CountyName COUNTY AUDITORS OFFIC', 
                                    '/CountyName COUNTY AUDITORS OFFICE', 
                                    '/CountyName COUNTY TREASURER', 
                                    '/CountyName County Treasurer', 
                                    '/CountyName COUNTY/ AUDITOR', 
                                    '/CountyName COUNTY/AUDITORS OFFIC', 
                                    '/CountyName COUNTY C/O AUDITORS') 
    AND AMOUNT &gt; 0


You may have your reasons for it, but there's a lot of duplication in your IN statement. Consider whether or not a LIKE...OR...IN type statement would meet your needs. It will perform worse, so test it against your original to see how bad the performance hit will be.

WHERE 
    CaseParticipantTable.FILING_NAME LIKE '/CountyName%COUNT%AUDITOR%'
    OR CaseParticipantTable.FILING_NAME IN(
                                    '/CountyName COUNTY TREASURER', 
                                    '/CountyName County Treasurer') 


This is an adhoc query now, but many many adhoc queries make their way into production use. Never should you Select * in a production query. It often forces a table scan where a more efficient query might be found otherwise.