How to SUM columns on multiple conditions in a GROUP BY

I'm not sure why you would need the multiple joins. Couldn't you simply do something like:

Select
    a.ACCOUNT_ID
    , a.BANK_NAME
    , a.LOCALE
    , a.STATUS
    , Sum ( t.Amount ) As Balance
    , Sum( Case When t.Amount < 0 Then Amount End ) As Outcome
    , Sum( Case When t.Amount > 0 Then Amount End ) As Income
From ACCOUNT a
    Left Join TRANSACTION t 
        On t.ACCOUNT_ID = a.ACCOUNT_ID
Group By a.ACCOUNT_ID, a.BANK_NAME, a.LOCALE, a.[STATUS]

Since you didn't tell us what's going wrong (that is, describe the behavior you get in addition to describing the the behavior you expect), it's hard to say where, but there are a couple of possibilities. Neil points out one. Another is that since you join on the transaction table three times, you're pairing transactions with transactions and getting repetitions. Instead, join on the transaction table a single time and change how you sum up the Amount column.

Select
    a.ACCOUNT_ID,
    a.BANK_NAME,
    a.LOCALE,
    a.STATUS,
    sum(t.AMOUNT) as BALANCE,
    sum((t.AMOUNT < 0) * t.AMOUNT) As OUTGOING,
    sum((t.AMOUNT > 0) * t.AMOUNT) As INCOMING
From ACCOUNT a
Left Join TRANSACTION t On t.ACCOUNT_ID = a.ACCOUNT_ID
Group By a.ACCOUNT_ID, a.BANK_NAME, a.LOCALE, a.[STATUS]

You can use CASE expressions as a more readable alternative to the multiplications:

Select
    a.ACCOUNT_ID,
    a.BANK_NAME,
    a.LOCALE,
    a.[STATUS],
    sum(t.AMOUNT) As BALANCE,
    sum(CASE WHEN t.AMOUNT < 0 THEN t.AMOUNT ELSE 0 end) As OUTCOME,
    sum(CASE WHEN t.AMOUNT > 0 THEN t.AMOUNT ELSE 0 end) As INCOME
From ACCOUNT a
Left Join [TRANSACTION] t On t.ACCOUNT_ID = a.ACCOUNT_ID
Group By a.ACCOUNT_ID, a.BANK_NAME, a.LOCALE, a.[STATUS]

Did you mean:

select 
    a.ACCOUNT_ID, 
    a.BANK_NAME, 
    a.LOCALE, 
    a.STATUS, 
    sum(t1.AMOUNT) as BALANCE, 
    sum(CASE WHEN t2.AMOUNT < 0 THEN t2.Amount ELSE 0 END) as OUTCOME, 
    sum(CASE WHEN t3.AMOUNT > 0 THEN t3.Amount ELSE 0 END) as INCOME 
from 
    ACCOUNT a 
    left join TRANSACTION t1 on t1.ACCOUNT_ID = a.ACCOUNT_ID 
    left join TRANSACTION t2 on t2.ACCOUNT_ID = a.ACCOUNT_ID
    left join TRANSACTION t3 on t3.ACCOUNT_ID = a.ACCOUNT_ID
group by 
    a.ACCOUNT_ID, a.BANK_NAME, a.LOCALE, a.[STATUS]