Use the unaccent module for that – which is completely different from what you are linking to.

unaccent is a text search dictionary that removes accents (diacritic
signs) from lexemes.

Install once per database with:

CREATE EXTENSION unaccent;

If you get an error like:

ERROR: could not open extension control file
"/usr/share/postgresql/<version>/extension/unaccent.control": No such file or directory

Install the contrib package on your database server like instructed in this related answer:

• Error when creating unaccent extension on PostgreSQL

Among other things, it provides the function unaccent() you can use with your example (where LIKE seems not needed).

SELECT *
FROM   users
WHERE  unaccent(name) = unaccent('João');

Index

To use an index for that kind of query, create an index on the expression. However, Postgres only accepts IMMUTABLE functions for indexes. If a function can return a different result for the same input, the index could silently break.

unaccent() only STABLE not IMMUTABLE

Unfortunately, unaccent() is only STABLE, not IMMUTABLE. According to this thread on pgsql-bugs, this is due to three reasons:

1. It depends on the behavior of a dictionary.
2. There is no hard-wired connection to this dictionary.
3. It therefore also depends on the current search_path, which can change easily.

Some tutorials on the web instruct to just alter the function volatility to IMMUTABLE. This brute-force method can break under certain conditions.

Others suggest a simple IMMUTABLE wrapper function (like I did myself in the past).

There is an ongoing debate whether to make the variant with two parameters IMMUTABLE which declares the used dictionary explicitly. Read here or here.

Another alternative would be this module with an IMMUTABLE unaccent() function by Musicbrainz, provided on Github. Haven’t tested it myself. I think I have come up with a better idea:

Best for now

This approach is more efficient as other solutions floating around, and safer.
Create an IMMUTABLE SQL wrapper function executing the two-parameter form with hard-wired schema-qualified function and dictionary.

Since nesting a non-immutable function would disable function inlining, base it on a copy of the C-function, (fake) declared IMMUTABLE as well. Its only purpose is to be used in the SQL function wrapper. Not meant to be used on its own.

The sophistication is needed as there is no way to hard-wire the dictionary in the declaration of the C function. (Would require to hack the C code itself.) The SQL wrapper function does that and allows both function inlining and expression indexes.

CREATE OR REPLACE FUNCTION public.immutable_unaccent(regdictionary, text)
  RETURNS text LANGUAGE c IMMUTABLE PARALLEL SAFE STRICT AS
'$libdir/unaccent', 'unaccent_dict';

CREATE OR REPLACE FUNCTION public.f_unaccent(text)
  RETURNS text LANGUAGE sql IMMUTABLE PARALLEL SAFE STRICT AS
$func$
SELECT public.immutable_unaccent(regdictionary 'public.unaccent', $1)
$func$;

Drop PARALLEL SAFE from both functions for Postgres 9.5 or older.

public being the schema where you installed the extension (public is the default).

The explicit type declaration (regdictionary) defends against hypothetical attacks with overloaded variants of the function by malicious users.

_{Previously, I advocated a wrapper function based on the STABLE function unaccent() shipped with the unaccent module. That disabled function inlining. This version executes ten times faster than the simple wrapper function I had here earlier.
And that was already twice as fast as the first version which added SET search_path = public, pg_temp to the function – until I discovered that the dictionary can be schema-qualified, too. Still (Postgres 12) not too obvious from documentation.}

If you lack the necessary privileges to create C functions, you are back to the second best implementation: An IMMUTABLE function wrapper around the STABLE unaccent() function provided by the module:

CREATE OR REPLACE FUNCTION public.f_unaccent(text)
  RETURNS text AS
$func$
SELECT public.unaccent('public.unaccent', $1)  -- schema-qualify function and dictionary
$func$  LANGUAGE sql IMMUTABLE PARALLEL SAFE STRICT;

Finally, the expression index to make queries fast:

CREATE INDEX users_unaccent_name_idx ON users(public.f_unaccent(name));

Remember to recreate indexes involving this function after any change to function or dictionary, like an in-place major release upgrade that would not recreate indexes. Recent major releases all had updates for the unaccent module.

Adapt queries to match the index (so the query planner will use it):

SELECT * FROM users
WHERE  f_unaccent(name) = f_unaccent('João');

You don’t need the function in the right expression. There you can also supply unaccented strings like 'Joao' directly.

The faster function does not translate to much faster queries using the expression index. That operates on pre-computed values and is very fast already. But index maintenance and queries not using the index benefit.

Security for client programs has been tightened with Postgres 10.3 / 9.6.8 etc. You need to schema-qualify function and dictionary name as demonstrated when used in any indexes. See:

• ‘text search dictionary “unaccent” does not exist’ entries in postgres log, supposedly during automatic analyze

Ligatures

In Postgres 9.5 or older ligatures like ‘Œ’ or ‘ß’ have to be expanded manually (if you need that), since unaccent() always substitutes a single letter:

SELECT unaccent('Œ Æ œ æ ß');

unaccent
----------
E A e a S

You will love this update to unaccent in Postgres 9.6:

Extend contrib/unaccent‘s standard unaccent.rules file to handle all
diacritics known to Unicode, and expand ligatures correctly (Thomas
Munro, Léonard Benedetti)

Bold emphasis mine. Now we get:

SELECT unaccent('Œ Æ œ æ ß');

unaccent
----------
OE AE oe ae ss

Pattern matching

For LIKE or ILIKE with arbitrary patterns, combine this with the module pg_trgm in PostgreSQL 9.1 or later. Create a trigram GIN (typically preferable) or GIST expression index. Example for GIN:

CREATE INDEX users_unaccent_name_trgm_idx ON users
USING gin (f_unaccent(name) gin_trgm_ops);

Can be used for queries like:

SELECT * FROM users
WHERE  f_unaccent(name) LIKE ('%' || f_unaccent('João') || '%');

GIN and GIST indexes are more expensive to maintain than plain btree:

• Difference between GiST and GIN index

There are simpler solutions for just left-anchored patterns. More about pattern matching and performance:

• Pattern matching with LIKE, SIMILAR TO or regular expressions in PostgreSQL

pg_trgm also provides useful operators for “similarity” (%) and “distance” (<->).

Trigram indexes also support simple regular expressions with ~ et al. and case insensitive pattern matching with ILIKE:

• PostgreSQL accent + case insensitive search