This can be done using group by and having clause, the condition is getting Claims having at least one Drug-X and none of H99 Code :

select t.*
from mytable t
inner join (
  select Claim_ID
  from mytable
  group by Claim_ID
  having count(case when Service = 'Drug-X' then 1 end) > 0
         and count(case when ICD10_Code = 'H99' then 1 end) = 0 
) as s on s.Claim_ID = t.Claim_ID

Demo here

It's probably more efficient to do this using window functions, rather than a self-join.

select t.*
from (
    select *,
      count_DrugX = count(case when Service = 'Drug-X' then 1 end) over (partition by Claim_ID),
      count_H99   = count(case when ICD10_Code = 'H99' then 1 end) over (partition by Claim_ID)
    from mytable t
) t
where count_DrugX > 0
  and count_H99 = 0;

Here's a solution that uses exists :

with yourData as (
    select *
    from (values
        ('123', '555', 'Injection', 'H99'),
        ('123', '555', 'Drug-X', NULL),
        ('123', '555', 'Exam',NULL),
        ('123', '555', 'Diagnosis', NULL),
        ('321', '556', 'Injection', 'H88'),
        ('321', '556', 'Drug-X', NULL),
        ('321', '556', 'Exam',  NULL),
        ('321', '556', 'Diagnosis',  NULL)
    ) as x(ClaimID, PersonID, [Service], [IDC10])
), badClaims as (
    select ClaimID
    from yourData
    where [Service] = 'Drug-X'
        and not exists (
            select *
            from yourData as t
            where yourData.ClaimID = t.ClaimID
                and IDC10 = 'H99'
        )
)
select *
from yourData
join badClaims
    on badClaims.ClaimID = yourData.ClaimID;

Breaking it down, the cte badClaims looks for line items that have Drug-X and no line item in the same claim has IDC10 = H99. Once such claims are found, we return all line items for that claim.