Can I draw something on my card and play tournaments with it?
You may not change or make illegible a card's name or mana cost. Other restrictions also apply. A tournament's head judge is the final authority on anything that happens in that tournament, including what card modifications are acceptable.


Yes you can obscure and remove the rules text.
You cannot completely replace the art work, the card must still be recognisable by its art.
The manacost must still be legible.
The name must still be legible.


The relevant rules for this question can be found in the MtG tournament rules (pdf):

http://wpn.wizards.com/sites/wpn/files/attachements/mtg_mtr_22jan16_en.pdf


  3.3 Authorized Cards
  
  Players may use any Authorized Game Cards from Magic: The Gathering expansions, core sets, special sets, supplements, and promotional printings. Authorized Game Cards are cards that, unaltered, meet the following conditions:
  
  
  The card is genuine and published by Wizards of the Coast
  The card has a standard Magic back or is a double-faced card.
  The card does not have squared corners.
  The card has black or white borders.
  The card is not a token card.
  The card is not damaged or modified in a way that might make it marked.
  The card is otherwise legal for the tournament as defined by the format.
  
  
  The Head Judge of an event may issue a proxy (see section 3.4) for a card that has become worn or damaged during the tournament.
  
  Any other cards that are not Authorized Game Cards are prohibited in all sanctioned tournaments.
  
  Unglued and Unhinged basic land cards are allowed in sanctioned Magic tournaments.
  
  Players may use cards from the Alpha printing only if the deck is in opaque sleeves.
  
  Players may use otherwise-legal non-English and/or misprinted cards provided they are not using them to create an advantage by using misleading text or pictures. Official promotional textless spells are allowed in sanctioned
  Magic tournaments in which they would otherwise be legal.
  
  Artistic modifications are acceptable in sanctioned tournaments, provided that the modifications do not make the card art unrecognizable, contain substantial strategic advice, or contain offensive images. Artistic modifications also may not obstruct or change the mana cost or name of the card.
  
  The Head Judge is the final authority on acceptable cards for a tournament


What constitutes "substantial strategic advice" is covered by 2.11:


  2.11 Taking Notes
  
  Players are allowed to take written notes during a match and may refer to those notes while that match is in progress. At the beginning of a match, each player’s note sheet must be empty and must remain visible throughout the match. Players do not have to explain or reveal notes to other players. Judges may ask to see a player’s notes and/or request that the player explain his or her notes.
  
  Players may not refer to other notes, including notes from previous matches, during games.
  
  Between games, players may refer to a brief set of notes made before the match. They are not required to reveal these notes to their opponents. These notes must be removed from the play area before the beginning of the next game. Excessive quantities of notes (more than a sheet or two) are not allowed and may be penalized as slow play.
  
  The use of electronic devices to take and refer to notes is permitted at Regular Rules Enforcement Level (see section 2.12 – Electronic Devices).
  
  Players and spectators (exception: authorized press) may not make notes while drafting. Players may not reference any outside notes during drafting, card pool registration, or deckbuilding. 
  
  Players may refer to Oracle text at any time. They must do so publicly and in a format which contains no other strategic information. Consulting online sources, such as gatherer.wizards.com, is allowed at Regular Rules Enforcement Level. If a player wishes to view Oracle text in private, he or she must ask a judge.
  
  Artistic modifications to cards that indirectly provide minor strategic information are acceptable. The Head Judge
  is the final arbiter on what cards and notes are acceptable for a tournament.