Calculate percentage of group using GROUP BY

You can do a self join against the total, using a dummy value as a key. For example:

SELECT
  t1.state AS state,
  t1.cnt AS cnt,
  100 * t1.cnt / t2.total as percent
FROM (
  SELECT
    state,
    COUNT(*) AS cnt,
    1 AS key
  FROM
    [publicdata:samples.natality]
  WHERE state is not null
  GROUP BY
    state) AS t1
JOIN (
  SELECT
    COUNT(*) AS total,
    1 AS key
  FROM
    [publicdata:samples.natality]) AS t2
ON t1.key = t2.key
ORDER BY percent DESC

You can use a window function to get the percentage of total by group, without the need for a subquery (improving on evan_b's solution):

SELECT 
   state
   ,count(*) / (sum(count(*)) OVER()) as pct
FROM  
   `bigquery-public-data.samples.natality` 
GROUP BY 
   state

Check ratio_to_report, one of the recently announced window functions:

SELECT state, ratio * 100 AS percent FROM (
 SELECT state, count(*) AS total, RATIO_TO_REPORT(total) OVER() AS ratio
 FROM [publicdata:samples.natality]
 GROUP by state
)

state   percent
AL      1.4201828131159113   
AK      0.23521048665998198  
AZ      1.3332896746620975   
AR      0.7709591206172346   
CA      10.008298605982642

Modifying Felipe's answer for the standard SQL BigQuery dialect instead of the Legacy SQL dialect looks like this:

select state, 100*(state_count / total) as pct
from (
  SELECT state, count(*) AS state_count, sum(count(*)) OVER() AS total
  FROM `bigquery-public-data.samples.natality` 
  GROUP by state
) s

Documentation of the standard SQL BigQuery aggregate analytic functions (aka 'window functions') is here: https://cloud.google.com/bigquery/docs/reference/standard-sql/analytic-function-concepts